#9788601

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

5018 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)(11,16)(12,14)(13,15)(17,18,23,25)(19,24,28,21)(20,26,30,32)(22,31,29,27);(1,2,4,6)(3,5,8,7)(9,10,13,15)(11,12,14,16)(17,19,23,28)(18,24,25,21)(20,29,30,22)(26,27,32,31);(1,2)(3,5)(4,6)(7,8)(18,25)(19,28)(20,30)(27,31);(9,11,13,14)(10,12,15,16)(17,20)(18,26)(19,29)(21,31)(22,28)(23,30)(24,27)(25,32);(1,2)(3,7)(4,6)(5,8)(9,10)(11,12)(13,15)(14,16)(17,21)(18,19)(20,31)(22,32)(23,24)(25,28)(26,29)(27,30);(1,3)(2,5)(4,8)(6,7)(9,12)(10,14)(11,15)(13,16)(17,22,23,29)(18,27,25,31)(19,20,28,30)(21,26,24,32)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=g,f^2=h,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, 9695797);