#9844071

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

4561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=gi,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, 9751267);