Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #15123 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,20)(21,24)(22,23);(1,2,3,4)(9,11)(10,13)(12,15)(14,16)(17,19,22,21)(18,20,23,24);(11,15)(13,16)(17,20,19,18,22,24,21,23);(2,4)(6,8)(18,23)(20,24)Polycyclic presentation:
2^9|a^2=gi,b^2=ei,c^2=fi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307727, #72307769, #72307868, … |
| maximal normal subgroup of | #72307727, #72307769, #72307868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341016);