Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,28)(24,26)(25,27);(1,2)(3,4)(14,19)(15,23)(16,18)(20,27)(24,25)(26,28);(6,8)(13,15)(14,20)(16,23)(17,24)(18,25)(19,26)(21,28)(22,27);(2,4)(10,12)(13,16,17,18)(14,21,19,22)(15,23,24,25)(20,28,26,27)Polycyclic presentation:
2^9|d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264942, #72265018, #72265067, … |
| maximal normal subgroup of | #72264942, #72265018, #72265067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324712);