#6393865

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×#3552	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3003 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)(9,10,13,18,14,17,22,31)(11,15,21,28,20,27,33,39)(12,16,24,29,23,30,25,32)(19,26,35,38,34,37,36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,26)(17,27)(18,28)(22,33)(23,34)(24,35)(25,36)(29,38)(30,37)(31,39)(32,40);(9,12,22,24,14,23,13,25)(10,16,18,32,17,30,31,29)(11,19,33,35,20,34,21,36)(15,26,28,40,27,37,39,38);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(11,20)(15,27)(19,34)(21,33)(26,37)(28,39)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,e^2=f,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

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, 6301061);