#6350771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2691 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)(7,13)(8,12)(10,15)(14,16)(18,24)(19,26)(21,22)(25,28)(27,32)(30,31);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,18)(19,21)(20,25)(22,24)(23,26)(27,30)(28,31)(29,32);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,13,12,16)(17,19,23,24)(18,21,26,22)(20,28,29,27)(25,31,32,30);(3,9)(7,14)(10,15)(13,16)(17,20)(18,25)(19,27)(21,30)(22,31)(23,29)(24,28)(26,32);(20,29)(25,32)(27,28)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257967);