#8943079

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

4726 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)(17,18,23,25)(19,22,30,24)(20,21,27,26)(28,29,32,31);(3,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,23,30)(18,24,25,22)(20,28,27,32)(21,31,26,29);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,25)(21,26)(22,24)(29,31);(11,16)(13,15)(17,20)(18,26)(19,28)(21,25)(22,31)(23,27)(24,29)(30,32);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,21,23,26)(18,27,25,20)(19,29,30,31)(22,32,24,28);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,22)(18,19)(20,31)(21,32)(23,24)(25,30)(26,28)(27,29)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8850275);