#6357544

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
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

1987 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)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18)(19,25)(20,30)(21,26)(22,31)(23,27)(24,28)(29,32);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,22,28,23,29,21,20)(18,24,26,25,27,30,31,32);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,12,13,16)(17,20,21,29,23,28,22,19)(18,25,31,24,27,32,26,30);(2,8)(6,13)(7,14)(12,16);(2,8)(3,9)(6,13)(10,15)(17,21,23,22)(18,26,27,31)(19,28,29,20)(24,32,30,25)

Polycyclic presentation:

2^9|b^2=f,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^b=eh,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, 6264740);