#6686919

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

1896 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)(15,20)(16,28)(17,22)(18,32)(19,23)(21,29)(24,38)(25,34)(26,35)(27,42)(30,36)(31,37)(33,40)(39,44)(41,43);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15)(14,20)(16,24)(17,26)(18,27)(19,25)(21,33)(22,35)(23,34)(28,38)(29,40)(30,39)(31,41)(32,42)(36,44)(37,43);(2,4)(13,16,18,29,19,30,17,31)(14,21,22,28,23,37,32,36)(15,24,27,40,25,39,26,41)(20,33,35,38,34,43,42,44);(5,7,9,8)(6,10,11,12)(15,25)(20,34)(24,39)(26,27)(33,43)(35,42)(38,44)(40,41);(2,4)(5,8,9,7)(6,10,11,12)(13,17,19,18)(14,22,23,32)(15,26,25,27)(16,29,30,31)(20,35,34,42)(21,36,37,28)(24,40,39,41)(33,44,43,38)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=f,e^2=fi,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, 6594115);