#506148

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	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

468 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)(20,32)(21,24)(22,35)(23,29)(25,28)(26,33)(30,39)(31,46)(34,47)(36,45)(37,48)(38,43)(40,42)(41,44);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,28,42,22,31,34,45,21,30,43,48,33,44,20,29)(18,23,32,41,26,37,38,39,24,36,47,46,35,40,25,27);(2,6)(5,7)(11,15)(13,16)(18,24)(23,36)(25,38)(26,35)(27,39)(32,47)(37,40)(41,46);(2,6)(5,7)(17,19,28,42,22,31,34,45,21,30,43,48,33,44,20,29)(18,23,32,41,26,37,38,39,24,36,47,46,35,40,25,27)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,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

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, 413344);