#6350850

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

2814 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^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, 6258046);