#9062584

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

4814 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969780);