#507299

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

544 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)(19,27)(20,24)(21,25)(22,26)(23,29)(28,41)(30,42)(31,43)(32,38)(33,39)(34,40)(35,44)(36,45)(37,46)(47,48);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,21,30,33,46,47,44,20,29,32,45,22,31,34,28)(18,23,25,36,39,43,48,41,24,27,38,42,26,37,40,35);(10,14)(13,15)(19,28)(21,34)(22,33)(23,35)(25,40)(26,39)(27,41)(29,44)(30,31)(32,47)(36,37)(38,48)(42,43)(45,46);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,24)(23,27)(25,38)(26,39)(35,41)(36,42)(37,43)(40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414495);