#6350829

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

2876 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)(3,14)(4,10)(5,11)(6,12)(7,13)(8,15)(9,16)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,3)(2,8)(4,5)(6,15)(7,16)(9,12)(10,13)(11,14)(20,27)(22,28)(26,31)(29,32);(1,4,7,15)(2,9,13,14)(3,5,16,6)(8,12,10,11)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(1,4)(2,10)(3,5)(6,16)(7,15)(8,13)(9,12)(11,14)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32);(1,5,7,6)(2,11,13,12)(3,4,16,15)(8,14,10,9)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

2^9|c^2=i,e^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, 6258025);