#6386397

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342800
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	C₂²×C₄	C₂⁵
prime radical	C₂×#3524	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2958 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)(3,7)(9,10,15,20,14,19,13,18)(11,21,23,33,24,16,28,31)(12,25,26,34,27,17,29,32)(22,30,38,40,37,35,36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,28)(14,24)(15,23)(17,30)(18,33)(19,21)(20,31)(25,35)(26,38)(27,37)(29,36)(32,40)(34,39);(9,12,14,27)(10,17,19,25)(11,22,24,37)(13,26,15,29)(16,30,21,35)(18,32,20,34)(23,36,28,38)(31,39,33,40);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,23,24,28)(12,26,27,29)(16,31,21,33)(17,32,25,34)(22,36,37,38)(30,39,35,40);(1,3,4,7)(2,6,5,8)(10,19)(11,24)(17,25)(18,20)(22,37)(23,28)(32,34)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293593);