#8997237

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

5272 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8904433);