#499535

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

Polycyclic presentation:

2^9|a^2=ei,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, 406731);