#417189

order	512
nilpotency class	3
isoclinism	isoclinic class of #397950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324385);