#9137502

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

5043 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)(11,12)(13,16)(14,15)(17,18,23,25)(19,28,30,24)(20,31,27,22)(21,26,32,29);(1,2)(3,4)(5,7)(6,8)(9,11,13,15)(10,12,16,14)(17,19,23,30)(18,24,25,28)(20,29,27,26)(21,31,32,22);(1,3,5,8)(2,6,7,4)(9,11)(10,14)(12,16)(13,15)(18,25)(20,27)(24,28)(26,29);(3,8)(4,6)(9,11,13,15)(10,12,16,14)(17,20,23,27)(18,22,25,31)(19,29,30,26)(21,28,32,24);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,21,23,32)(18,26,25,29)(19,22,30,31)(20,24,27,28);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,16)(13,14)(17,22)(18,27)(19,21)(20,25)(23,31)(24,26)(28,29)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9044698);