#9181831

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

5095 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=h,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, 9089027);