#9065594

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

4882 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972790);