#9250307

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

5170 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=h,e^2=h,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, 9157503);