#503929

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	#24778	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,22)(19,24,29,37)(20,25,32,38)(21,26,36,35)(27,39,44,43)(28,33,41,46)(30,40,47,45)(31,34,42,48);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19)(18,24)(20,30)(21,33)(22,37)(23,29)(25,40)(26,41)(27,31)(28,35)(32,47)(34,39)(36,46)(38,45)(42,44)(43,48);(9,11,12,16)(10,14,13,15)(17,20,23,32)(18,25,22,38)(19,27,29,44)(21,34,36,48)(24,39,37,43)(26,42,35,31)(28,45,41,40)(30,46,47,33);(2,6)(5,7)(10,13)(14,15)(17,18,23,22)(19,24,29,37)(20,25,32,38)(21,26,36,35)(27,39,44,43)(28,33,41,46)(30,40,47,45)(31,34,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=fi,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411125);