#412613

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

864 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)(3,6)(4,5)(7,8)(9,10,13,18,15,19,14,29)(11,16,22,20,24,32,23,31)(12,25,26,33,28,38,27,17)(21,34,35,40,37,39,36,30);(2,5)(6,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,30)(18,20)(19,32)(25,34)(26,35)(27,36)(28,37)(29,31)(33,40)(38,39);(1,3,4,7)(2,6,5,8)(9,10)(11,20)(12,17)(13,29)(14,18)(15,19)(16,22)(21,34)(23,32)(24,31)(25,27)(26,38)(28,33)(30,35)(36,40)(37,39);(9,12,14,27,15,28,13,26)(10,17,18,25,19,33,29,38)(11,21,23,36,24,37,22,35)(16,30,20,34,32,40,31,39)

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,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, 319809);