Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #52163 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,27)(20,25)(21,26)(22,23)(24,28)(29,31)(30,32);(1,2)(3,7)(4,8)(5,6)(10,14)(11,16);(9,11)(10,12)(13,16)(14,15)(18,23)(19,28)(25,32)(26,29);(9,12)(10,11)(13,15)(14,16)(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,30)(31,32);(1,3,5,4)(2,7,6,8)(17,20)(18,25)(19,29)(21,24)(22,30)(23,32)(26,28)(27,31);(1,4,5,3)(2,8,6,7)(17,21)(18,26)(19,25)(20,27)(22,31)(23,29)(24,30)(28,32)Polycyclic presentation:
2^9|e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72303576 |
| maximal normal subgroup of | #72303576 |
| maximal quotient of | #72303659 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566264);