#7642419

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

4981 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=gh,f^2=g,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, 7549615);