#9137652

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

5328 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,e^2=h,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, 9044848);