#507324

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 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)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,33)(21,37)(22,28)(23,29)(24,27)(26,35)(30,40)(31,41)(32,42)(34,45)(36,46)(38,48)(39,43)(44,47);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,30,38,21,35,39,36,22,31,23,32,24,20,34,47)(18,25,40,48,37,26,43,46,28,41,29,42,27,33,45,44);(10,14)(13,15)(17,20,22,35)(18,26,28,33)(19,21,31,24)(23,38,30,32)(25,27,41,37)(29,42,40,48)(34,47,39,36)(43,44,45,46);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,21,22,24)(18,27,28,37)(19,20,31,35)(23,34,30,39)(25,26,41,33)(29,43,40,45)(32,36,38,47)(42,44,48,46)

Polycyclic presentation:

2^9|b^2=fg,c^2=i,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gh,h^c=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, 414520);