#507328

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

Polycyclic presentation:

2^9|b^2=fgi,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, 414524);