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 | |
8119 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)(6,10)(7,11)(14,19)(15,24)(22,28)(23,25);(5,6,9,10)(7,8,11,12)(13,14,18,19)(15,21,24,20)(16,22,26,28)(17,23,27,25);(2,3)(5,7)(6,8)(9,11)(10,12)(13,15,18,24)(14,20,19,21)(16,25,26,23)(17,22,27,28);(1,3,4,2)(5,8)(6,11)(7,10)(9,12)(13,15)(14,21)(16,25)(17,22)(18,24)(19,20)(23,26)(27,28);(13,16)(14,22)(15,25)(17,21)(18,26)(19,28)(20,27)(23,24);(13,17)(14,23)(15,22)(16,20)(18,27)(19,25)(21,26)(24,28)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7604499);