#6353329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353301
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

2890 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,22,18)(20,26,30,24)(21,27,23,25)(28,31,32,29);(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,21,22,23)(18,25,19,27)(20,29,30,31)(24,32,26,28);(3,10)(7,14)(11,16)(13,15)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338423, #72338425, #72338447
maximal normal subgroup of	#72338423, #72338425, #72338447

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260525);