#504111

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

565 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=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, 411307);