#387898

order	512
nilpotency class	3
isoclinism	isoclinic class of #232138
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

888 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)(3,5)(4,6)(7,8)(9,10,16,19,15,21,14,20)(11,22,25,17,26,34,31,35)(12,18,27,37,28,38,13,29)(23,39,40,33,30,36,24,32);(1,3,5,2)(4,7,8,6)(9,11)(10,17)(12,24)(13,30)(14,31)(15,26)(16,25)(18,36)(19,34)(20,22)(21,35)(23,27)(28,40)(29,33)(32,37)(38,39);(1,4,5,8)(2,6,3,7)(9,12,14,27,15,28,16,13)(10,18,20,37,21,38,19,29)(11,23,25,24,26,30,31,40)(17,32,34,36,35,33,22,39);(2,3)(6,7)(9,13,16,28,15,27,14,12)(10,18,20,37,21,38,19,29)(11,24,31,23,26,40,25,30)(17,33,34,39,35,32,22,36)

Polycyclic presentation:

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

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, 295094);