#6459911

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1729 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

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

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, 6367107);