#507938

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(17,18,21,26)(19,23,28,36)(20,24,33,40)(22,34,32,27)(25,42,41,31)(29,39,43,35)(30,47,44,38)(37,45,48,46);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,29)(21,28)(22,30)(24,39)(25,37)(26,36)(27,38)(31,46)(32,44)(33,43)(34,47)(35,40)(41,48)(42,45);(10,13)(14,16)(18,24)(19,28)(20,31)(22,32)(23,35)(25,27)(26,40)(29,45)(33,42)(34,41)(36,39)(37,47)(38,48)(43,46);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,28)(23,36)(29,43)(30,44)(35,39)(37,48)(38,47)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=egi,d^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

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, 415134);