Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1257 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1257 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(6,9)(7,11)(10,12)(17,20)(22,25)(23,27)(26,28);(1,2)(3,4)(5,6,8,10)(7,12,11,9)(13,14)(15,17)(16,18)(19,20);(2,4)(13,15)(14,17)(16,19)(18,20)(21,22,24,26)(23,28,27,25)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cf,d^b=dg,e^b=eh,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72217697, #72279344, #72279393 |
| maximal normal subgroup of | #72217697, #72279344, #72279393 |
| maximal quotient of | #72214372, #72214507, #72216703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10108);