#9781237

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

4775 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=h,e^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, 9688433);