#507380

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,21,27,20,26,34,45,23,29,37,46,22,28,36,38)(19,24,31,40,33,44,48,43,32,41,47,42,30,39,35,25);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,31)(22,30)(23,32)(25,38)(26,44)(27,40)(28,39)(29,41)(34,48)(35,36)(37,47)(42,46)(43,45);(11,15)(13,16)(17,19)(18,25)(20,30)(21,35)(22,33)(23,32)(24,38)(26,42)(27,39)(28,40)(29,43)(31,36)(34,47)(37,48)(41,45)(44,46);(2,6)(5,7)(17,20,23,22)(18,26,29,28)(19,30,32,33)(21,34,37,36)(24,39,41,44)(25,42,43,40)(27,45,46,38)(31,35,47,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,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, 414576);