#507988

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

456 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egi,d^2=g,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, 415184);