#509231

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25777	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

747 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)(15,24)(16,19)(17,32)(18,23)(20,26)(21,28)(22,30)(25,40)(27,43)(29,39)(31,35)(33,38)(34,42)(36,41)(37,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(15,25)(20,36)(24,40)(26,41)(27,33)(29,44)(37,39)(38,43);(2,4)(14,19)(16,28)(17,30)(20,24)(21,22)(23,35)(26,29)(27,33)(31,42)(32,34)(36,40)(37,43)(38,39)(41,44);(6,10)(7,12)(13,15,18,25)(14,20,23,36)(16,29,31,44)(17,33,30,27)(19,24,35,40)(21,38,34,43)(22,39,32,37)(26,42,41,28)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416427);