#6458890

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,24)(21,31)(22,25)(23,29)(27,28)(30,32);(9,11)(10,13)(12,15)(14,16)(17,19,21,29,22,30,20,28)(18,23,24,26,25,27,31,32);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,18)(19,27)(20,31)(21,24)(22,25)(23,30)(26,28)(29,32);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,18)(19,23)(20,31)(21,24)(22,25)(26,29)(27,30)(28,32);(17,20,22,21)(18,24,25,31)(19,28,30,29)(23,26,27,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366086);