#7643261

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

4750 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550457);