#72238232

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 15875 conjugacy classes of subgroups; only the 31743 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 28 points:

(1,2,3,4)(5,6,9,10)(7,8,11,12)(15,24)(19,20)(25,28)(26,27);(1,2)(3,4)(6,10)(7,11)(13,14)(15,19)(16,21)(17,22)(18,23)(20,24)(25,27)(26,28);(5,7,9,11)(6,8,10,12)(13,15,18,24)(14,19,23,20)(16,25,22,28)(17,26,21,27);(1,2)(3,4)(6,10)(7,11)(13,15,18,24)(14,19,23,20)(16,25,22,28)(17,26,21,27);(1,2)(3,4)(5,8)(6,11)(7,10)(9,12)(13,15)(14,20)(16,25)(17,27)(18,24)(19,23)(21,26)(22,28);(5,6)(7,8)(9,10)(11,12)(13,16)(14,21)(15,25)(17,23)(18,22)(19,27)(20,26)(24,28);(5,8)(6,11)(7,10)(9,12)(13,17,18,21)(14,22,23,16)(15,26,24,27)(19,28,20,25)

Polycyclic presentation:

2^10|a^2=h,c^2=hij,d^2=i,g^2=i,b^a=bh,c^a=ci,c^b=chj,d^a=dhi,d^c=dhj,e^a=eij,e^b=ei,e^c=ei,e^d=ei,f^a=fhj,f^b=fhj,f^d=fhj,f^e=fhj,g^e=gi,g^f=ghij

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238707
maximal normal subgroup of	#72238707

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);