#412614

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=hi,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, 319810);