#72328752

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 9572 conjugacy classes of subgroups; only the 3349 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gj,c^2=g,f^2=ij,b^a=bgh,c^b=cghi,d^a=di,d^b=di,d^c=di,e^a=egj,e^b=egj,e^c=egj,f^a=fij,f^b=fgh,f^c=fgh,f^d=fgh,f^e=fghij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328873
maximal normal subgroup of	#72328873

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