#8898080

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

4909 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:

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

Polycyclic presentation:

2^9|c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805276);