#72328299

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 6963 conjugacy classes of subgroups; only the 3453 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)(7,10)(8,9)(11,12)(13,14)(15,17)(16,19)(18,20)(21,22,25,26)(23,24,28,27);(1,2)(3,4)(6,8)(10,12)(13,14)(15,18)(16,19)(17,20)(22,26)(24,27);(13,15)(14,17)(16,20)(18,19);(6,8)(10,12)(21,23)(22,27)(24,26)(25,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,28)(26,27);(5,8,9,6)(7,10,11,12)(13,14)(15,17)(16,19)(18,20)(21,24)(22,28)(23,26)(25,27)

Polycyclic presentation:

2^10|a^2=h,f^2=ij,b^a=bhi,c^b=cg,d^a=dhij,e^a=eg,f^a=fij,f^b=fhij,f^d=fij,f^e=fghij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328461
maximal normal subgroup of	#72328461
maximal quotient of	#72331076, #72331185

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