#6367711

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367534
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

1585 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:

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

Polycyclic presentation:

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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274907);