#507498

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

498 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eg,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414694);