Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2213 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)(7,11)(8,9)(10,12);(1,2)(3,4)(6,9)(10,11)(13,14)(15,20)(16,25)(17,22)(18,27)(19,23)(21,26)(24,28);(2,4)(13,15,19,16)(14,20,23,25)(17,26,18,24)(21,27,28,22);(2,4)(13,16,19,15)(14,21,23,28)(17,26,18,24)(20,27,25,22);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,22,23,27)(15,24,16,26)(20,28,25,21)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72284843, #72307631, #72307862, … |
| maximal normal subgroup of | #72284843, #72307631, #72307862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425647);