#6370943

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4089 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22,24,16)(13,17)(14,18)(19,33,35,27)(20,28)(21,29)(23,36,38,30)(25,31)(26,32)(34,40,37,39);(2,5)(6,8)(9,11,14,21)(10,15,18,29)(12,19,24,35)(13,25,26,20)(16,27,22,33)(17,31,32,28)(23,37,38,34)(30,40,36,39);(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,22)(20,34)(21,35)(25,37)(26,38)(28,39)(29,33)(31,40)(32,36);(1,3,4,7)(2,6,5,8);(9,13,14,26)(10,17,18,32)(11,20,21,25)(12,23,24,38)(15,28,29,31)(16,30,22,36)(19,34,35,37)(27,39,33,40)

Polycyclic presentation:

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

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, 6278139);