#501350

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

703 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)(7,10)(8,9)(11,12);(6,9)(10,12)(13,14,18,22)(15,23,24,41)(16,27,31,43)(17,32,29,21)(19,34,35,25)(20,37,39,30)(26,42,33,38)(28,40,44,36);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,28)(17,33)(18,24)(20,38)(21,40)(22,35)(23,27)(25,30)(26,29)(31,44)(32,36)(34,37)(39,42)(41,43);(6,9)(10,12)(15,24)(19,35)(23,41)(25,34)(26,33)(28,44)(36,40)(38,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,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, 408546);