#391281

order	512
nilpotency class	3
isoclinism	isoclinic class of #219525
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1025 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)(3,5)(4,6)(7,8)(9,10,15,21,16,20,14,19)(11,22,13,17,25,32,24,31)(12,18,26,35,27,36,29,37)(23,34,28,30,38,40,39,33);(1,3)(2,5)(4,7)(6,8)(9,11,16,25)(10,17,20,31)(12,23,27,38)(13,15,24,14)(18,33,36,30)(19,32,21,22)(26,39,29,28)(34,35,40,37);(2,6)(5,8)(9,12)(10,18)(11,23)(13,28)(14,29)(15,26)(16,27)(17,30)(19,37)(20,36)(21,35)(22,34)(24,39)(25,38)(31,33)(32,40);(3,7)(5,8)(9,13,14,11,16,24,15,25)(10,17,19,22,20,31,21,32)(12,23,26,28,27,38,29,39)(18,34,35,30,36,40,37,33)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298477);