#6348116

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi

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, 6255312);