#6461393

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447056
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2572
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=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, 6368589);