#424930

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
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

1204 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=f,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=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, 332126);