#6683095

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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	C₂×#3054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590291);