#6351147

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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

2792 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:

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

Polycyclic presentation:

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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258343);