Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418488 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,12,18,28)(8,21,23,30)(9,16)(10,15)(11,14)(13,24,31,25)(17,33)(19,27,20,32)(22,36)(26,35)(29,34);(5,7,11,19,9,18,10,20)(6,12,14,27,16,28,15,32)(8,22,24,17,23,35,25,34)(13,29,21,36,31,33,30,26);(2,4)(5,8,9,23)(6,13)(7,17,18,34)(10,24,11,25)(12,26)(14,30)(15,21)(16,31)(19,22,20,35)(27,33)(28,36)(29,32);(6,14,16,15)(7,18)(12,27,28,32)(13,30,31,21)(17,34)(19,20)(22,35)(26,33,36,29)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=e,d^2=ei,f^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^b=de,f^b=fi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318258, #72318867, #72318947 |
| maximal normal subgroup of | #72318258, #72318867, #72318947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348415);