#9075971

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

5024 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)(17,18,23,24)(19,21,20,25)(22,30,31,26)(27,32,28,29);(1,2,5,6)(3,4,7,8)(11,15)(14,16)(18,24)(21,25)(26,30)(29,32);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,25,24,21)(22,27,31,28)(26,32,30,29);(3,7)(4,8)(17,20,23,19)(18,21,24,25)(22,27,31,28)(26,32,30,29);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,20)(19,24)(22,29)(23,25)(26,28)(27,30)(31,32);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=i,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, 8983167);