#9075992

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

4844 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:

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

Polycyclic presentation:

2^9|b^2=gi,e^2=i,f^2=g,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, 8983188);