#6364165

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1534 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271361);