#505577

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,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, 412773);