#503645

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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

673 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)(17,32)(18,21)(19,36)(20,27)(22,30)(23,35)(24,26)(28,42)(31,40)(33,37)(34,41)(38,44)(39,43);(1,2)(3,4)(13,15,18,28)(14,20,21,37)(16,19,31,26)(17,30,34,44)(22,36,38,24)(23,40,39,29)(25,41,42,32)(27,43,33,35);(2,4)(5,7,8,12)(6,9,10,11)(14,21)(20,37)(22,38)(23,39)(24,36)(25,42)(29,40)(32,41);(13,16,18,31)(14,22,21,38)(15,26,28,19)(17,33,34,27)(20,24,37,36)(23,25,39,42)(29,41,40,32)(30,43,44,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=i,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,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410841);