#9271791

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

4992 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=ghi,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, 9178987);