#9730336

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

4899 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)(3,4)(5,7)(6,8)(10,14)(11,16)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32);(9,11,13,16)(10,15,14,12)(18,25)(19,28)(20,30)(26,32);(9,12)(10,16)(11,14)(13,15)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31);(1,4,5,6)(2,3,7,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=gh,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, 9637532);