#7655165

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

4780 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:

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

Polycyclic presentation:

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

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, 7562361);