Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | |
2189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,19)(15,20,24,28)(16,25,23,21)(17,22,27,26);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,18,24)(14,20,19,28)(16,26,23,22)(17,21,27,25);(6,10)(7,11)(13,16)(14,21)(15,22)(17,20)(18,23)(19,25)(24,26)(27,28);(6,10)(9,12)(13,17,14,22,18,27,19,26)(15,23,28,25,24,16,20,21);(1,3,4,2)(16,23)(17,27)(21,25)(22,26)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257078, #72257096, #72257163, … |
| maximal normal subgroup of | #72257078, #72257096, #72257163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518736);