#507752

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 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)(17,18,20,23)(19,27,28,45)(21,33,31,46)(22,36,32,26)(24,40,38,29)(25,42,39,35)(30,47,37,43)(34,44,48,41);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(18,23)(24,38)(25,39)(26,36)(27,45)(33,46)(41,44)(43,47);(10,13)(14,16)(17,19,20,28)(18,24,23,38)(21,34,31,48)(22,37,32,30)(25,43,39,47)(26,44,36,41)(27,46,45,33)(29,42,40,35);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,23)(24,38)(25,39)(26,36)(27,45)(33,46)(41,44)(43,47)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,d^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,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, 414948);