#72238190

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 25226 conjugacy classes of subgroups; only the 31739 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:

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

Polycyclic presentation:

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

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