#507326

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

535 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)(19,23)(20,25)(21,26)(22,24)(27,34)(28,35)(29,38)(30,36)(31,37)(32,40)(33,39)(41,45)(42,46)(43,47)(44,48);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,27,41,33,44,32,28,22,29,20,30,21,31,42,43)(18,23,34,45,39,48,40,35,24,38,25,36,26,37,46,47);(10,14)(13,15)(19,28)(20,32)(21,33)(23,35)(25,40)(26,39)(27,42)(29,43)(30,31)(34,46)(36,37)(38,47)(41,44)(45,48);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,24)(19,29)(25,34)(26,39)(28,43)(30,41)(31,44)(40,46)

Polycyclic presentation:

2^9|b^2=fgi,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, 414522);