#416444

order	512
nilpotency class	3
isoclinism	isoclinic class of #396613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
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

946 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)(11,19)(12,16)(13,25)(14,17)(15,21)(18,27)(20,33)(22,32)(23,36)(24,30)(26,29)(28,34)(31,38)(35,40)(37,39);(9,11)(10,15)(12,20,24,35)(13,26)(14,22)(16,28,30,39)(17,29)(18,32)(19,25)(21,27)(23,37,38,34)(31,40,36,33);(2,5)(6,7)(9,12,13,23,14,24,27,38)(10,16,18,31,17,30,25,36)(11,20,21,34,22,35,26,37)(15,28,19,33,29,39,32,40);(1,3,4,8)(2,6,5,7)(10,17)(15,29)(16,30)(18,25)(19,32)(28,39)(31,36)(33,40)

Polycyclic presentation:

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

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, 323640);