#9788739

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

4578 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)(11,12)(13,15)(14,16)(17,18,22,24)(19,23,27,31)(20,25,30,29)(21,28,32,26);(1,2,4,6)(3,5,8,7)(9,11,14,15)(10,13,16,12)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,32)(26,30);(1,2)(3,5)(4,6)(7,8)(18,24)(19,27)(25,29)(26,28);(9,12,14,13)(10,15,16,11)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,32)(26,30);(1,2)(3,7)(4,6)(5,8)(9,13,14,12)(10,15,16,11)(17,20)(18,25)(19,28)(21,31)(22,30)(23,32)(24,29)(26,27);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,21,22,32)(18,26,24,28)(19,29,27,25)(20,23,30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=g,e^2=g,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, 9695935);