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 | |
8316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(5,6,8,10)(7,9,12,11)(13,14,18,16)(15,20,23,19)(17,24,25,21)(22,26,28,27);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,20)(17,22)(18,23)(21,26)(24,27)(25,28);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,23)(14,19,16,20)(17,22,25,28)(21,26,24,27);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,19)(17,22)(18,23)(21,27)(24,26)(25,28);(13,16,18,14)(15,19,23,20)(17,24,25,21)(22,26,28,27);(13,17)(14,21)(15,22)(16,24)(18,25)(19,26)(20,27)(23,28)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=gh,e^2=i,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, 7585158);