#507915

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=fh,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415111);