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 | |
7430 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,26)(17,22)(18,20)(21,23)(24,25)(27,28);(1,2,3,4)(13,15)(14,19)(16,24)(17,27)(18,23)(20,21)(22,28)(25,26);(5,6,8,10)(7,9,12,11)(14,20)(15,23)(16,22)(25,27);(5,6)(7,9)(8,10)(11,12)(13,15)(14,21)(16,24)(17,25)(18,23)(19,20)(22,28)(26,27);(5,7)(6,9)(8,12)(10,11)(13,16)(14,17)(15,24)(18,22)(19,27)(20,26)(21,25)(23,28);(13,17)(14,22)(15,25)(16,20)(18,26)(19,24)(21,28)(23,27)Polycyclic presentation:
2^9|b^2=g,c^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7879031);