#6350426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350397
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

2844 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,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,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257622);