#6366774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366710
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

1548 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)(3,4)(5,10)(6,8)(7,11)(9,12)(13,16)(14,15)(18,24)(19,26)(20,28)(21,22)(25,29)(27,32);(2,8)(3,12)(4,14)(5,6)(9,16)(10,11)(17,18)(19,23)(20,29)(21,26)(22,24)(25,31)(27,30)(28,32);(1,3)(2,9)(4,11)(5,15)(6,13)(7,12)(8,14)(10,16)(17,19,23,18)(20,27,28,25)(21,26,22,24)(29,31,32,30);(1,4,7,9)(2,3,11,12)(5,14,6,16)(8,13,10,15)(17,20,23,28)(18,25,19,27)(21,30,22,31)(24,32,26,29);(1,5,7,6)(2,8,11,10)(3,13,12,15)(4,14,9,16)(17,21,23,22)(18,26,19,24)(20,30,28,31)(25,29,27,32)

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6273970);