#72329631

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6682 conjugacy classes of subgroups; only the 3377 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=h,e^2=ghi,f^2=h,b^a=bhj,c^b=cg,d^a=dhi,d^b=dhi,e^a=egi,e^b=eghi,e^d=ehi,f^d=fh,f^e=fgi

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329703
maximal normal subgroup of	#72329703
maximal quotient of	#72331230

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