#503999

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

Aliases

No aliases for this group.

Subgroups

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

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=eh,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, 411195);