#6808384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1802 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715580);