#423050

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1108 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)(9,10,14,18)(11,19,22,16)(12,15,27,33)(13,28,30,26)(17,31,36,32)(20,24,25,23)(21,38,29,37)(34,39,35,40);(1,3,4,8)(2,5,6,7)(9,11,14,22)(10,16,18,19)(12,25,27,20)(13,29,30,21)(15,24,33,23)(17,35,36,34)(26,37,28,38)(31,39,32,40);(9,12,15,18,14,27,33,10)(11,20,23,19,22,25,24,16)(13,26,31,36,30,28,32,17)(21,37,39,35,29,38,40,34);(3,8)(5,7)(9,13,14,30)(10,17,18,36)(11,21,22,29)(12,26,27,28)(15,31,33,32)(16,34,19,35)(20,37,25,38)(23,39,24,40)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=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, 330246);