#503766

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	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=f,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,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, 410962);