#6459000

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,26)(21,24)(22,27)(23,29)(28,32)(30,31);(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,27,32)(20,29,21,28)(24,25,26,31);(2,5)(6,8)(11,15)(13,16)(17,20,22,21)(18,24,27,26)(19,28,30,29)(23,31,32,25);(11,15)(13,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,24)(26,32)(27,31);(1,3,4,7)(2,6,5,8)(17,20,22,21)(18,26,27,24)(19,29,30,28)(23,31,32,25)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=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, 6366196);