#501327

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

714 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)(7,11)(8,10)(9,12)(13,14,20,34,16,23,33,37,19,26,17,24,18,25,31,44)(15,21,35,41,30,43,32,22,29,40,28,39,27,38,36,42);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,30)(17,28)(18,27)(19,29)(20,35)(22,37)(23,43)(24,39)(25,38)(26,40)(31,36)(32,33)(34,41)(42,44);(2,4)(13,15)(14,22)(16,27)(17,32)(18,30)(19,29)(20,36)(21,37)(23,41)(24,38)(25,39)(26,42)(28,33)(31,35)(34,43)(40,44);(6,10)(9,11)(13,16,19,18)(14,23,26,25)(15,27,29,30)(17,31,20,33)(21,38,40,43)(22,41,42,39)(24,44,34,37)(28,32,35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408523);