#9713369

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

4803 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)(11,12)(13,16)(14,15)(17,18)(19,28)(20,26)(21,27)(22,24)(23,25)(29,32)(30,31);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,14)(15,16)(17,19,23,22)(18,24,25,28)(20,29,30,27)(21,31,32,26);(9,10,15,14)(11,13,16,12)(18,25)(20,30)(24,28)(27,29);(9,12,15,13)(10,16,14,11)(17,20,23,30)(18,26,25,31)(19,29,22,27)(21,28,32,24);(1,4,5,3)(2,6,7,8)(9,13,15,12)(10,16,14,11)(17,21)(18,27)(19,26)(20,28)(22,31)(23,32)(24,30)(25,29);(1,2)(3,6)(4,8)(5,7)(9,14,15,10)(11,13,16,12)(17,22,23,19)(18,28,25,24)(20,27,30,29)(21,26,32,31)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=gh,f^2=h,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, 9620565);