#6385999

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×#3500	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^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, 6293195);