Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2011 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10)(11,12)(14,18)(15,23)(16,19)(20,22)(21,28)(24,26);(5,6,7,8)(13,14,19,20,17,22,16,18)(15,21,26,27,24,28,23,25);(2,4)(6,8)(14,20)(15,23)(16,19)(18,22)(24,26)(25,27);(10,12)(13,15,19,26,17,24,16,23)(14,21,20,27,22,28,18,25)Polycyclic presentation:
2^9|b^2=gi,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264385, #72264400, #72264438, … |
| maximal normal subgroup of | #72264385, #72264400, #72264438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371395);