Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3460 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(13,14)(15,22)(16,20)(17,25)(18,21)(19,24)(23,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,26)(18,23)(20,28)(21,27)(22,25);(6,9)(10,12)(15,23)(16,17)(19,27)(20,25);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)Polycyclic presentation:
2^9|a^2=h,d^2=g,e^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279210 |
| maximal normal subgroup of | #72279210 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266065);