#6792418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6792416
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

1827 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:

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

Polycyclic presentation:

2^9|e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,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, 6699614);