#8946462

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

4845 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=i,e^2=g,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, 8853658);