#6386000

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×#3499	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4042 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)(4,6)(5,8)(9,10,14,20,15,19,13,18)(11,16,22,30,23,31,27,33)(12,24,25,34,26,17,28,32)(21,35,37,39,38,29,36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,27)(14,22)(15,23)(17,29)(18,33)(19,31)(20,30)(24,35)(25,37)(26,38)(28,36)(32,40)(34,39);(9,12,15,26)(10,17,19,24)(11,21,23,38)(13,25,14,28)(16,29,31,35)(18,32,20,34)(22,36,27,37)(30,39,33,40);(3,8)(5,7)(9,11)(10,16)(12,21)(13,27)(14,22)(15,23)(17,29)(18,33)(19,31)(20,30)(24,35)(25,37)(26,38)(28,36)(32,40)(34,39);(2,6)(5,7)(9,13,15,14)(10,18,19,20)(11,22,23,27)(12,25,26,28)(16,30,31,33)(17,32,24,34)(21,36,38,37)(29,39,35,40)

Polycyclic presentation:

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

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, 6293196);