#506109

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=fhi,d^2=ehi,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=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, 413305);