#508614

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

469 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,43)(30,44)(31,42)(32,41)(33,38)(34,45)(35,36)(37,40)(39,46)(47,48);(2,5)(6,8)(17,19,20,29)(18,23,24,27)(21,34,32,43)(22,37,33,31)(25,28,41,45)(26,42,38,40)(30,47,46,36)(35,39,48,44);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,18)(19,28)(20,24)(21,35)(22,38)(23,34)(25,36)(26,33)(27,43)(29,45)(30,40)(31,39)(32,48)(37,44)(41,47)(42,46);(1,3,4,7)(2,6,5,8)(18,24)(23,27)(25,41)(26,38)(28,45)(35,48)(39,44)(40,42)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415810);