#499488

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

593 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)(13,14,19,25,17,23,32,36,18,24,33,43,20,26,35,39)(15,27,29,40,30,37,34,22,16,21,31,38,28,42,44,41);(1,2)(3,4)(6,9)(10,12)(13,15)(14,21)(16,18)(17,30)(19,29)(20,28)(22,39)(23,42)(24,27)(25,38)(26,37)(31,33)(32,34)(35,44)(36,41)(40,43);(2,4)(13,16,18,15)(14,22,24,41)(17,30,20,28)(19,34,33,44)(21,36,27,39)(23,40,26,38)(25,37,43,42)(29,35,31,32);(5,7,8,11)(6,10,9,12)(13,17,18,20)(14,23,24,26)(15,28,16,30)(19,32,33,35)(21,37,27,42)(22,40,41,38)(25,36,43,39)(29,44,31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406684);