#72304584

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 25489 conjugacy classes of subgroups; only the 31843 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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=i,d^2=j,g^2=i,c^a=ci,c^b=cj,d^a=di,d^b=di,d^c=dj,e^c=eij,e^d=eij,f^c=fh,f^d=fh,g^a=gi,g^b=gi,g^c=gj,g^e=gi,g^f=gi

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304735
maximal normal subgroup of	#72304735

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