Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2351 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)(13,14,16,21,17,20,24,19)(15,18,23,26,22,27,28,25);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,27)(21,26)(24,28);(6,9)(10,12)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(5,7)(6,10)(8,11)(9,12)(14,20)(15,22)(19,21)(23,28);(7,11)(10,12)Polycyclic presentation:
2^9|a^2=g,b^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72284610, #72284948, #72307846, … |
| maximal normal subgroup of | #72284610, #72284948, #72307846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452201);