Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 ◁ #2398 ◁ 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)(15,24)(16,25)(17,26)(18,23)(19,20)(21,22)(27,28);(2,4)(6,10)(11,12)(13,15)(14,20)(16,26)(18,19)(21,28)(23,24);(5,7,9,8)(6,11,10,12)(13,16,15,22,18,26,19,27)(14,21,20,25,23,28,24,17);(1,2,3,4)(13,17,19,28,18,25,15,21)(14,22,24,16,23,27,20,26);(5,8,9,7)(6,11,10,12)(14,23)(16,26)(20,24)(22,27)Polycyclic presentation:
2^9|c^2=fgi,d^2=h,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,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257048, #72257096, #72257213, … |
| maximal normal subgroup of | #72257048, #72257096, #72257213, … |
| maximal quotient of | #72264342, #72264591 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521700);