#72238191

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 13512 conjugacy classes of subgroups; only the 31773 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=h,g^2=ij,b^a=bh,c^b=chi,d^a=dhi,d^c=dhi,e^a=ei,e^c=ei,f^a=fhi,f^c=fhi,f^e=fhi,g^a=gi,g^b=gij,g^c=gi,g^d=gij,g^e=gij,g^f=gi

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, 2, F.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9);;
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);