#8806947

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

4852 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)(18,19)(20,30)(22,31)(25,28);(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,19)(20,24,30,27)(21,28,32,25)(22,29,31,26);(1,2)(3,6)(4,5)(7,8)(10,13)(14,15)(17,19,23,18)(20,24,30,27)(21,25,32,28)(22,29,31,26);(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,31)(22,32)(23,30)(25,29)(26,28);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,32)(18,25,19,28)(20,31,30,22)(24,29,27,26);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,14,13,15)(17,22,23,31)(18,26,19,29)(20,32,30,21)(24,28,27,25)

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714143);