#6393551

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342866
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×#3551	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3180 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)(9,10,15,19,14,20,13,26)(11,18,22,30,23,16,27,31)(12,17,24,35,25,34,28,33)(21,32,38,40,37,29,36,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,23)(10,16,20,18)(12,21,25,37)(13,27,15,22)(17,29,34,32)(19,31,26,30)(24,38,28,36)(33,40,35,39);(9,12,15,28,14,25,13,24)(10,17,26,35,20,34,19,33)(11,21,22,36,23,37,27,38)(16,29,30,39,18,32,31,40);(3,8)(5,7)(9,11)(10,18)(12,21)(13,27)(14,23)(15,22)(16,20)(17,32)(19,30)(24,38)(25,37)(26,31)(28,36)(29,34)(33,39)(35,40);(2,6)(5,7)(9,13,14,15)(10,19,20,26)(11,22,23,27)(12,24,25,28)(16,30,18,31)(17,33,34,35)(21,36,37,38)(29,39,32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,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, 6300747);