#505070

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

427 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)(9,10,13,16)(11,15,12,14)(17,18)(19,28)(20,33)(21,26)(22,27)(23,25)(24,31)(29,44)(30,39)(32,45)(34,48)(35,43)(36,42)(37,41)(38,47)(40,46);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,30)(21,31)(22,29)(23,37)(25,39)(26,28)(27,38)(32,36)(33,41)(34,46)(35,47)(40,42)(43,44)(45,48);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,15,16,14)(19,29)(20,23)(22,36)(24,38)(25,33)(27,42)(28,44)(30,32)(31,47)(34,35)(37,46)(39,45)(40,41)(43,48);(1,4,5,3)(2,7,8,6)(10,16)(14,15)(17,20,21,23)(18,25,26,33)(19,30,31,37)(22,34,35,36)(24,39,28,41)(27,42,43,48)(29,46,47,32)(38,40,44,45)

Polycyclic presentation:

2^9|a^2=ei,b^2=ei,c^2=ei,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,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, 412266);