#504466

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

632 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)(13,14,20,35,34,43,33,38,19,26,17,24,18,25,37,23)(15,27,30,39,31,41,32,21,16,22,28,44,29,42,36,40);(1,2)(3,4)(13,15,19,16)(14,21,26,40)(17,32,20,36)(18,31,34,29)(22,38,27,23)(24,42,35,41)(25,44,43,39)(28,37,30,33);(2,4)(6,9)(10,12)(13,16,19,15)(14,22,26,27)(17,30,20,28)(18,31,34,29)(21,38,40,23)(24,39,35,44)(25,41,43,42)(32,33,36,37);(5,7,8,11)(6,10,9,12)(14,23)(17,33)(18,34)(20,37)(21,22)(24,43)(25,35)(26,38)(27,40)(28,32)(29,31)(30,36)(39,42)(41,44)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=i,d^2=f,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, 411662);