Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25731 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,19,25,27)(15,21,23,26,22,28,24,18);(2,3)(13,15)(14,18)(16,24)(17,22)(19,26)(20,28)(21,27)(23,25);(1,2,4,3)(6,9)(10,12)(14,19)(15,22)(20,27)(23,24);(1,3,4,2)(14,19)(15,22)(20,27)(23,24);(5,7,8,11)(6,10,9,12)(14,19)(18,26)(20,27)(21,28)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307564, #72307590, #72307986, … |
| maximal normal subgroup of | #72307564, #72307590, #72307986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374573);