#9788892

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

4520 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9696088);