Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10390 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
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 | |
1472 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1472 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(13,14)(15,21)(16,19,24,27)(17,20)(18,23)(22,26,28,25);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18,20,21)(16,22)(17,23)(19,26,27,25)(24,28);(2,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,19,20,27)(15,22)(17,24)(18,25,21,26)(23,28)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=ei,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72210067, #72210069, #72251457, … |
| maximal normal subgroup of | #72210067, #72210069, #72251457, … |
| maximal quotient of | #72210066, #72255990, #72258293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29234);