Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2458 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)(15,22,23,18)(17,19,25,21)(24,27,28,26);(1,2)(3,4)(6,9)(10,12)(13,15,16,23)(14,18,20,22)(17,24,25,28)(19,26,21,27);(2,4)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(15,23)(18,22)(24,28)(26,27);(2,4)(14,20)(15,23)(19,21)(24,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282968, #72283580, #72283919, … |
| maximal normal subgroup of | #72282968, #72283580, #72283919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482968);