#499534

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

822 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,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, 406730);