#9720351

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9627547);