#8611919

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

5769 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8519115);