#512345

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

529 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efi,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419541);