#72328465

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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 9860 conjugacy classes of subgroups; only the 3301 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)(7,11)(8,9)(10,12)(13,14)(15,18)(16,20)(17,19)(21,22)(23,24)(25,26)(27,28);(13,14)(15,20)(16,18)(17,19)(21,23)(22,27)(24,25)(26,28);(1,2,4,3)(6,9)(10,11)(13,15,17,16)(14,18,19,20)(21,24)(22,28)(23,25)(26,27);(1,2,4,3)(5,6,8,9)(7,10,12,11)(13,14)(15,20)(16,18)(17,19)(21,24)(22,28)(23,25)(26,27);(2,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,18)(16,20)(17,19)(21,23)(22,24)(25,27)(26,28);(1,3,4,2)(5,7,8,12)(6,10,9,11)(13,16,17,15)(14,18,19,20)(21,25,26,22)(23,24,28,27)

Polycyclic presentation:

2^10|c^2=gh,d^2=ghij,f^2=h,b^a=bi,c^a=cg,c^b=cgj,d^a=dg,d^c=dij,e^c=eghj,e^d=eghj,f^a=fj,f^c=fi,f^d=fi,f^e=fh

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328649
maximal normal subgroup of	#72328649

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