#6792445

order	512
nilpotency class	3
isoclinism	isoclinic class of #6792444
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#38904
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6699641);