#503989

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

584 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)(13,14,21,28)(15,29,33,23)(16,22,35,43)(17,36,32,42)(18,38,30,26)(19,40,37,25)(20,41,39,27)(24,34,44,31);(1,2)(3,4)(6,9)(10,11)(13,15,19,31,20,32,18,16,21,33,37,34,39,17,30,35)(14,22,26,42,27,24,25,29,28,43,38,36,41,44,40,23);(2,4)(13,16,30,32,39,31,37,15,21,35,18,17,20,34,19,33)(14,23,40,44,41,36,38,43,28,29,25,24,27,42,26,22);(5,7,8,12)(6,10,9,11)(13,17,37,16,20,15,30,34,21,32,19,35,39,33,18,31)(14,24,38,23,27,43,40,42,28,44,26,29,41,22,25,36)

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

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, 411185);