#6357510

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357480
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2065 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264706);