#499410

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

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,g^c=gh,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, 406606);