#418734

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325930);