#72236855

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 15855 conjugacy classes of subgroups; only the 31775 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:

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

Polycyclic presentation:

2^10|a^2=h,c^2=h,d^2=i,g^2=j,b^a=bhi,c^b=ch,d^a=dh,d^c=dh,e^b=ei,e^d=ei,f^b=fh,f^d=fh,g^a=gi,g^b=gij,g^c=gi,g^d=gij,g^e=gj,g^f=ghi

Conjugacy classes & character table

184 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);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);