#410442

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

827 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)(3,6)(4,5)(7,8)(10,16)(11,21)(12,26)(13,14)(17,30)(19,20)(24,25)(27,28)(29,33)(31,36)(35,37)(39,40);(9,10,14,19,15,20,13,16)(11,22,24,17,25,32,21,30)(12,18,26,33,28,34,27,29)(23,36,37,31,38,40,35,39);(2,5)(6,8)(9,11,14,24,15,25,13,21)(10,17,16,22,20,30,19,32)(12,23,26,37,28,38,27,35)(18,31,29,36,34,39,33,40);(1,3,4,7)(2,6,5,8)(9,12,13,27,15,28,14,26)(10,18,16,29,20,34,19,33)(11,23,21,35,25,38,24,37)(17,31,22,36,30,39,32,40)

Polycyclic presentation:

2^9|b^2=h,c^2=g,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,g^a=gi,h^a=hi,c^b=cg,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, 317638);