#505573

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,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, 412769);