Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23428 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,20)(16,24)(17,26)(18,21)(19,23)(22,27)(25,28);(7,8)(10,12)(13,15)(14,19)(16,18)(17,25)(20,26)(21,28)(22,24)(23,27);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16,18,15)(14,20,21,24)(17,25,27,23)(19,22,28,26);(5,8,9,7)(6,10,11,12)(15,16)(19,28)(20,24)(23,25);(14,21)(19,28)(20,24)(22,26)Polycyclic presentation:
2^9|c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249631, #72249662, #72281161, … |
| maximal normal subgroup of | #72249631, #72249662, #72281161, … |
| maximal quotient of | #72264933, #72308459 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370054);