#9778481

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

4952 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)(11,15)(12,13)(14,16)(17,18)(19,28)(20,30)(21,25)(22,29)(23,24)(26,31)(27,32);(1,2,7,8)(3,4,6,5)(10,13)(14,15)(18,24)(19,21)(26,30)(27,29);(1,3,7,6)(2,5,8,4)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,25,24,28)(20,27,31,29)(22,26,32,30);(1,4,7,5)(2,3,8,6)(9,11)(10,14)(12,16)(13,15)(17,20,23,31)(18,26,24,30)(19,29,21,27)(22,28,32,25);(1,5,7,4)(2,3,8,6)(17,21,23,19)(18,25,24,28)(20,27,31,29)(22,30,32,26);(1,6,7,3)(2,5,8,4)(17,22)(18,27)(19,26)(20,28)(21,30)(23,32)(24,29)(25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9685677);