#412976

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

948 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,14)(13,26)(15,19)(16,20)(17,23)(18,28)(22,36)(24,32)(25,31)(27,34)(29,33)(30,39)(35,37)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,21)(13,22)(14,24)(15,23)(16,25)(18,30)(19,31)(20,32)(26,35)(27,38)(28,37)(29,39)(33,40)(34,36);(1,4,5,3)(2,6,7,8)(9,12,14,10,16,19,15,20)(11,21,23,32,25,31,24,17)(13,26,27,18,29,33,28,34)(22,35,37,36,39,40,38,30);(9,13,14,27,16,29,15,28)(10,18,19,33,20,34,12,26)(11,22,24,38,25,39,23,37)(17,30,31,40,32,36,21,35)

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=hi,e^2=i,f^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,e^b=ei,f^b=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, 320172);