#6352478

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352469
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259674);