#505579

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
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	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

649 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)(19,25)(20,32)(21,27)(22,36)(23,40)(24,29)(26,34)(28,38)(30,41)(31,45)(33,43)(35,46)(37,44)(39,48)(42,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,24,20)(18,25,29,32)(21,33,23,30)(22,31,39,34)(26,36,45,48)(27,43,40,41)(28,42,46,44)(35,37,38,47);(11,15)(13,16)(17,20,24,19)(18,26,29,45)(21,33,23,30)(22,37,39,47)(25,36,32,48)(27,44,40,42)(28,41,46,43)(31,38,34,35);(2,6)(5,7)(17,21,24,23)(18,27,29,40)(19,30,20,33)(22,35,39,38)(25,41,32,43)(26,44,45,42)(28,36,46,48)(31,47,34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412775);