Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,16,20,17,21,25,18)(15,22,24,26,23,19,27,28);(1,2)(3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(14,18)(16,25)(19,26)(20,21)(22,28)(24,27);(2,4)(6,9)(10,12)(13,15,17,23)(14,19,21,22)(16,24,25,27)(18,26,20,28);(5,7,8,11)(6,10,9,12)(15,23)(19,22)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72307396, #72307628, #72307905, … |
| maximal normal subgroup of | #72307396, #72307628, #72307905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7055413);