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 | |
8114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(6,7)(10,11)(13,14,18,19)(15,17,21,26)(16,20,24,27)(22,25,23,28);(5,6,9,7)(8,10,12,11)(13,15)(14,17)(16,23)(18,21)(19,26)(20,28)(22,24)(25,27);(2,3)(5,7,9,6)(8,10,12,11)(14,19)(15,21)(16,24)(25,28);(1,3,4,2)(5,8,9,12)(6,10,7,11)(13,15)(14,17)(16,22)(18,21)(19,26)(20,25)(23,24)(27,28);(13,16)(14,20)(15,22)(17,25)(18,24)(19,27)(21,23)(26,28);(13,17)(14,15)(16,25)(18,26)(19,21)(20,22)(23,27)(24,28)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=hi,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7583428);