#504182

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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)(9,10,12,13)(11,15,16,14);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,18,23,27)(19,28,31,46)(20,25,22,37)(21,33,36,47)(24,39,42,30)(26,32,44,35)(29,45,38,43)(34,40,48,41);(9,11,12,16)(10,14,13,15)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,40)(26,43)(27,42)(28,33)(30,35)(32,39)(36,48)(37,41)(44,45)(46,47);(2,6)(5,7)(10,13)(14,15)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,40,42,41)(26,33,44,47)(28,45,46,43)(30,34,39,48)

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=f,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, 411378);