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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 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)(13,14,20,22)(15,18,17,19)(16,23,24,27)(21,26,28,25);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,21)(17,20)(19,22)(23,26)(24,28)(25,27);(5,7,8,11)(6,10,9,12)(13,16,19,26,20,24,18,25)(14,21,17,27,22,28,15,23);(2,4)(13,17,18,14,20,15,19,22)(16,23,25,28,24,27,26,21);(6,9)(10,12)(16,24)(21,28)(23,27)(25,26)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=g,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=cgi,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 | #72308392, #72308510, #72308732, … |
| maximal normal subgroup of | #72308392, #72308510, #72308732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468407);