#6364893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6364762
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

1611 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,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, 6272089);