#511312

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=efi,c^2=ef,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,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, 418508);