#8843614

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

4934 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)(9,10,12,14)(11,15,16,13)(17,18,23,26)(19,22,29,24)(20,30,25,21)(27,28,32,31);(3,7)(6,8)(9,10,12,14)(11,15,16,13)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,25)(18,21,26,30)(19,27,29,32)(22,31,24,28);(9,10,12,14)(11,13,16,15)(17,21,23,30)(18,25,26,20)(19,28,29,31)(22,27,24,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,26)(24,29)(25,30)(27,31)(28,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,22)(18,19)(20,31)(21,27)(23,24)(25,28)(26,29)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8750810);