#9788960

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

4773 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)(17,18,22,27)(19,23,30,24)(20,25,31,29)(21,28,32,26);(1,2,4,6)(3,5,8,7)(10,13)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,27,23)(20,28,31,26)(21,29,32,25);(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31);(1,2)(3,7)(4,6)(5,8)(17,20)(18,25)(19,28)(21,24)(22,31)(23,32)(26,30)(27,29);(1,3)(2,5)(4,8)(6,7)(17,21,22,32)(18,26,27,28)(19,29,30,25)(20,23,31,24)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9696156);