#504280

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

534 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)(13,14,19,25)(15,27,31,22)(16,33,29,21)(17,24,34,38)(18,23,28,41)(20,37,36,26)(30,43,35,40)(32,42,44,39);(1,2)(3,4)(6,9)(10,11)(13,15,18,30,20,32,34,16,19,31,28,35,36,44,17,29)(14,21,24,39,26,40,41,22,25,33,38,42,37,43,23,27);(2,4)(13,16,34,44,36,30,18,31,19,29,17,32,20,35,28,15)(14,22,41,43,37,39,24,33,25,27,23,40,26,42,38,21);(5,7,8,12)(6,10,9,11)(13,15,28,35,20,32,17,29,19,31,18,30,36,44,34,16)(14,21,38,42,26,40,23,27,25,33,24,39,37,43,41,22)

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=fh,d^2=ehi,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, 411476);