#9183918

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

4913 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=gh,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, 9091114);