Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,8)(10,11)(13,14,19,22)(15,23,26,20)(16,25,27,21)(17,28,24,18);(1,2,3,4)(13,15,19,26)(14,20,22,23)(16,24,27,17)(18,25,28,21);(5,6,9,8)(7,11,12,10)(13,16,19,27)(14,21,22,25)(15,17,26,24)(18,20,28,23);(5,7,9,12)(6,10,8,11)(13,16)(14,21)(15,24)(17,26)(18,20)(19,27)(22,25)(23,28);(5,8,9,6)(7,11,12,10)(13,17,19,24)(14,18,22,28)(15,16,26,27)(20,25,23,21);(13,18)(14,17)(15,25)(16,20)(19,28)(21,26)(22,24)(23,27)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72280781 |
| maximal normal subgroup of | #72280781 |
| maximal quotient of | #72280903, #72281083, #72281269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7804853);