Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(2,4)(14,18)(15,22)(16,19)(20,27)(21,25)(23,26);(6,9)(10,12)(13,14,19,25,17,21,16,18)(15,20,26,28,23,27,22,24);(13,15,19,26,17,23,16,22)(14,20,25,28,21,27,18,24);(5,7,8,11)(6,10,9,12)(14,21)(18,25)(20,27)(24,28)Polycyclic presentation:
2^9|c^2=hi,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307078, #72307082, #72307624, … |
| maximal normal subgroup of | #72307078, #72307082, #72307624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374472);