Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2033 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)(7,11)(8,12)(9,10)(13,14)(15,23)(16,24)(17,19)(18,26)(20,21)(22,25)(27,28);(1,2)(3,4)(6,10)(11,12)(14,19)(15,21)(16,20)(18,25)(23,27)(24,28);(5,7,9,8)(6,11,10,12)(13,15,18,21,17,24,25,28)(14,20,22,23,19,27,26,16);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,20,17,23,25,27)(14,21,22,24,19,28,26,15);(5,8,9,7)(6,11,10,12)(14,19)(15,24)(21,28)(22,26)Polycyclic presentation:
2^9|c^2=fgi,d^2=fgi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257549, #72257625, #72263869, … |
| maximal normal subgroup of | #72257549, #72257625, #72263869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472645);