#505207

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

716 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)(13,14,20,27)(15,29,32,44)(16,34,35,24)(17,37,33,23)(18,26,39,43)(19,25,38,41)(21,40,30,28)(22,36,42,31);(1,2)(3,4)(6,9)(10,12)(13,15,19,16,21,33,39,36,20,32,38,35,30,17,18,31)(14,22,26,37,28,24,41,44,27,42,43,23,40,34,25,29);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,15,30,31,38,17,20,35,39,32,21,36,19,33)(14,23,25,42,40,44,43,24,27,37,41,22,28,29,26,34);(6,9)(10,12)(13,17,38,36,21,15,18,35,20,33,19,31,30,32,39,16)(14,24,43,29,28,42,25,37,27,34,26,44,40,22,41,23)

Polycyclic presentation:

2^9|a^2=i,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,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, 412403);