#385862

order	512
nilpotency class	3
isoclinism	isoclinic class of #220210
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1388 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)(3,4)(5,7)(6,8)(9,10,13,19,14,18,26,34)(11,15,22,32,23,17,21,31)(12,24,28,36,29,16,27,37)(20,35,25,40,38,30,39,33);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,25)(13,22)(14,23)(16,33)(17,18)(19,32)(20,27)(21,26)(24,40)(28,38)(29,39)(30,36)(31,34)(35,37);(1,4,5,6)(2,3,7,8)(9,12,26,28,14,29,13,27)(10,16,34,37,18,24,19,36)(11,20,22,39,23,38,21,25)(15,30,32,40,17,35,31,33);(2,7)(3,8)(9,11)(10,17)(12,20)(13,21)(14,23)(15,18)(16,35)(19,32)(22,26)(24,30)(25,27)(28,39)(29,38)(31,34)(33,37)(36,40)

Polycyclic presentation:

2^9|a^2=h,c^2=egi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,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, 293058);