#6366984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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

1575 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)(3,13)(4,12)(5,10)(6,9)(7,11)(8,14)(15,16)(18,24)(19,26)(21,22)(25,32)(27,28)(29,31);(1,3,7,15)(2,8,11,5)(4,14,6,10)(9,13,12,16)(17,18,23,19)(20,28,30,32)(21,26,22,24)(25,31,27,29);(1,4,7,6)(2,9,11,12)(3,14,15,10)(5,16,8,13)(17,19,23,18)(20,27,30,25)(21,26,22,24)(28,29,32,31);(1,5)(2,10)(3,9)(4,16)(6,13)(7,8)(11,14)(12,15)(17,20,23,30)(18,25,19,27)(21,31,22,29)(24,32,26,28);(2,11)(3,15)(9,12)(10,14)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,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=ehi,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, 6274180);