#506178

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	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

459 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)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,33)(21,24)(22,36)(23,31)(25,30)(26,34)(28,40)(29,42)(32,48)(35,37)(38,45)(39,44)(41,47)(43,46);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,28,44,22,32,20,31,21,29,30,47,34,46,35,45)(18,23,37,48,26,39,25,27,24,38,33,43,36,41,40,42);(2,6)(5,7)(11,15)(13,16)(18,24)(19,29)(25,40)(26,36)(31,45)(32,46)(33,37)(44,47);(2,6)(5,7)(17,19,30,47,22,32,35,45,21,29,28,44,34,46,20,31)(18,23,33,43,26,39,40,42,24,38,37,48,36,41,25,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=ehi,d^2=efhi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,d^c=di

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, 413374);