#72328069

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
prime radical	#648360	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

22478 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|e^2=i,f^2=hi,b^a=bj,c^b=cg,e^b=eh,e^d=egi,f^a=fh,f^d=fg

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328076
maximal normal subgroup of	#72328076
maximal quotient of	#72331630

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