#9333321

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

4784 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)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,23,20)(19,24,26,21)(22,25,31,28)(27,32,30,29);(1,3)(2,8)(4,6)(5,7)(10,13)(14,15)(17,19,23,26)(18,24,20,21)(22,30,31,27)(25,29,28,32);(1,2,7,6)(3,8,5,4)(9,10)(11,14)(12,13)(15,16)(19,26)(21,24)(27,30)(29,32);(1,4)(2,5)(3,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,18)(19,21,26,24)(22,28,31,25)(27,29,30,32);(1,5,7,3)(2,8,6,4)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,30)(31,32);(1,6,7,2)(3,8,5,4)(17,22,23,31)(18,25,20,28)(19,27,26,30)(21,29,24,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9240517);