#6372228

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342612
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,d^2=i,e^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279424);