Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10390 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1268 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1268 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6)(7,8,10,12)(9,11)(13,14)(15,21,23,18)(16,24)(17,20)(19,26)(22,27,25,28);(1,2)(3,4)(5,7)(6,8,11,12)(9,10)(13,15)(14,18)(16,25,26,22)(17,23)(19,28,24,27)(20,21);(2,4)(5,8)(6,10,11,7)(9,12)(13,16)(14,19,20,24)(15,22)(17,26)(18,27,21,28)(23,25)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,f^b=fh,g^b=gh,e^c=eh,f^c=fi,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72210069, #72251435, #72251436, … |
| maximal normal subgroup of | #72210069, #72251435, #72251436, … |
| maximal quotient of | #72210062, #72255799, #72255826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27829);