#6357989

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357957
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1851 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)(17,18,23,27)(19,21,25,28)(20,30,26,22)(24,31,32,29);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,22,29,23,25,30,31)(18,24,26,21,27,32,20,28);(17,20,22,18,23,26,30,27)(19,28,29,24,25,21,31,32);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,21)(18,25)(19,27)(20,29)(22,32)(23,28)(24,30)(26,31);(3,9)(7,14)(10,15)(12,16)(19,25)(21,28)(24,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265185);