#6379253

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342728
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3298 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286449);