#6350358

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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

2658 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)(18,19)(20,28)(21,22)(24,31)(25,26)(27,32);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(17,18,23,26)(19,21,25,22)(20,24,28,32)(27,29,31,30);(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,31)(30,32);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,20,23,28)(18,24,26,32)(19,27,25,31)(21,29,22,30);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(20,28)(24,32)(27,31)(29,30)

Polycyclic presentation:

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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257554);