#9063253

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

5205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=ghi,f^2=h,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, 8970449);