#504465

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

643 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)(7,10)(8,9)(11,12)(13,14,19,33,31,40,30,39,17,21,16,23,18,24,35,22)(15,25,43,38,32,44,29,20,27,37,26,42,28,41,34,36);(1,2)(3,4)(13,15,17,27)(14,20,21,36)(16,29,19,34)(18,32,31,28)(22,37,39,25)(23,41,33,44)(24,42,40,38)(26,35,43,30);(2,4)(6,9)(10,12)(14,21)(20,36)(22,39)(23,33)(24,40)(25,37)(38,42)(41,44);(5,7,8,11)(6,10,9,12)(14,22)(16,30)(18,31)(19,35)(20,37)(21,39)(23,40)(24,33)(25,36)(26,29)(28,32)(34,43)(38,41)(42,44)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,d^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 411661);