Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,12)(8,13,22,31)(9,16)(10,14)(11,15)(17,25,33,36)(18,27)(19,26)(20,28)(21,35,34,29)(23,30,24,32);(1,2)(3,4)(5,7)(6,12)(8,21,22,34)(9,19)(10,20)(11,18)(13,29,31,35)(14,28)(15,27)(16,26)(17,23,33,24)(25,32,36,30);(2,4)(5,8)(6,13,15,31)(7,17)(9,24)(10,23)(11,22)(12,25,27,36)(14,30,16,32)(18,33)(19,34)(20,21)(26,29,28,35);(6,14,15,16)(7,18)(12,26,27,28)(13,30,31,32)(17,33)(19,20)(21,34)(25,29,36,35)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280175, #72280622, #72280664, … |
| maximal normal subgroup of | #72280175, #72280622, #72280664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308675);