#72232623

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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 10391 conjugacy classes of subgroups; only the 3329 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:

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

Polycyclic presentation:

2^10|c^2=hi,d^2=ij,e^2=ghi,c^a=cg,c^b=ch,d^a=dghi,d^c=dij,e^a=eghi,e^c=eghi,e^d=eghj,f^b=fghi,f^c=fhij,f^d=fj,f^e=fhij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232676
maximal normal subgroup of	#72232676

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