#511514

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

466 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)(10,13)(11,15)(17,18,22,24)(19,23,30,36)(20,25,33,39)(21,26,32,40)(27,35,44,46)(28,37,43,47)(29,38,34,42)(31,41,45,48);(1,3,6,8)(2,4,7,5)(9,10,12,13)(11,14,15,16)(17,19,22,30)(18,23,24,36)(20,27,33,44)(21,34,32,29)(25,35,39,46)(26,42,40,38)(28,45,43,31)(37,48,47,41);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16)(19,27)(20,31)(21,32)(23,35)(25,41)(26,40)(28,29)(30,44)(33,45)(34,43)(36,46)(37,38)(39,48)(42,47);(1,5,6,4)(2,3,7,8)(10,13)(14,16)(18,24)(23,36)(25,39)(26,40)(35,46)(37,47)(38,42)(41,48)

Polycyclic presentation:

2^9|a^2=efi,b^2=ei,c^2=e,d^2=ef,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, 418710);