#407566

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314762);