Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(13,14,17,21,18,15,22,20)(16,19,25,27,24,23,28,26);(5,6,8,10)(7,11,12,9)(13,15)(14,18)(16,23)(17,20)(19,24)(21,22)(25,26)(27,28);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,19)(15,23)(17,25)(18,24)(20,26)(21,27)(22,28);(16,24)(19,23)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238811, #72238813, #72238814, … |
| maximal normal subgroup of | #72238811, #72238813, #72238814, … |
| maximal quotient of | C2×#358779, #72238827, #72238829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265975);