#504270

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

540 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:

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

Polycyclic presentation:

2^9|a^2=efi,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, 411466);