#6686924

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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	#30512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1899 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594120);