#512334

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

602 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)(7,9)(8,10)(11,12)(13,14,18,22)(15,23,27,39)(16,21,30,36)(17,25,33,41)(19,24,34,40)(20,35,32,26)(28,42,43,38)(29,44,31,37);(5,7,8,11)(6,9,10,12)(13,15,19,28,20,29,33,30,18,27,34,43,32,31,17,16)(14,21,25,37,26,38,40,39,22,36,41,44,35,42,24,23);(6,10)(9,12)(14,22)(15,27)(16,30)(24,40)(25,41)(26,35)(28,43)(29,31);(2,4)(13,16,17,31,32,43,34,27,18,30,33,29,20,28,19,15)(14,23,24,42,35,44,41,36,22,39,40,38,26,37,25,21)

Polycyclic presentation:

2^9|a^2=i,b^2=efhi,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=ghi,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419530);