#6462881

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2414 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370077);