#9189442

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

4915 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)(3,8)(4,7)(5,6)(9,10,12,14)(11,13,15,16)(19,26)(20,28)(22,23)(24,29);(1,2,6,5)(3,7,4,8)(11,15)(13,16)(17,18,21,25)(19,23,26,22)(20,24,28,29)(27,31,32,30);(1,3,6,4)(2,7,5,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,26)(18,22,25,23)(20,27,28,32)(24,30,29,31);(3,4)(7,8)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30);(1,4,6,3)(2,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,31)(30,32);(1,5,6,2)(3,8,4,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,29)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9096638);