#9015671

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

5502 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922867);