#499431

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

618 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)(7,12)(8,10)(9,11)(13,14,17,23,16,22,30,41,19,25,33,42,18,24,32,34)(15,20,27,36,29,40,44,39,28,37,43,38,26,35,31,21);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,29)(17,27)(18,26)(19,28)(21,34)(22,40)(23,36)(24,35)(25,37)(30,44)(31,32)(33,43)(38,42)(39,41);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(14,21)(16,26)(17,31)(18,29)(19,28)(20,34)(22,38)(23,35)(24,36)(25,39)(27,32)(30,43)(33,44)(37,41)(40,42);(13,16,19,18)(14,22,25,24)(15,26,28,29)(17,30,33,32)(20,35,37,40)(21,38,39,36)(23,41,42,34)(27,31,43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=i,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, 406627);