#8657173

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

4685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gi,e^2=hi,f^2=gi,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, 8564369);