#505195

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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×#135	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

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=fi,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

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, 412391);