#6552624

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,3)(13,14,17,32,18,23,19,22)(15,24,27,36,28,20,26,35)(16,29,33,21,31,37,30,38)(25,40,41,39,43,44,42,34);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,34)(22,36)(23,24)(29,44)(30,42)(31,43)(32,35)(33,41)(37,40)(38,39);(1,3,4,2)(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,31)(19,33)(20,34)(22,37)(23,38)(24,39)(26,41)(27,42)(28,43)(29,32)(35,44)(36,40);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,32)(15,26,28,27)(16,30,31,33)(20,35,24,36)(21,37,38,29)(25,41,43,42)(34,44,39,40);(7,11)(10,12)(13,17,18,19)(14,22,23,32)(15,27,28,26)(16,30,31,33)(20,36,24,35)(21,37,38,29)(25,42,43,41)(34,40,39,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459820);