#9064080

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

4788 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,e^2=g,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, 8971276);