#6442153

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1895 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349349);