#501794

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	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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)(15,19,24,32)(16,20,29,37)(17,21,28,36)(23,31,39,42)(25,33,30,38)(26,34,41,44)(27,35,40,43);(2,4)(5,7,8,11)(6,9,10,12)(15,23)(16,26)(17,28)(19,31)(20,34)(21,36)(24,39)(25,40)(27,30)(29,41)(32,42)(33,43)(35,38)(37,44);(6,10)(9,12)(13,15)(14,19)(16,27)(17,30)(18,24)(20,35)(21,38)(22,32)(23,26)(25,28)(29,40)(31,34)(33,36)(37,43)(39,41)(42,44);(2,4)(15,24)(19,32)(23,39)(25,30)(27,40)(31,42)(33,38)(35,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 408990);