#500122

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3280	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407318);