Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1009 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)(8,19,21,32)(9,14)(10,24)(11,15)(13,22,31,23)(16,33,34,30)(17,27)(18,29)(20,36,35,26)(25,28);(5,7,9,17,11,18,10,25)(6,12,14,27,15,29,24,28)(8,20,23,16,21,35,22,34)(13,30,32,36,31,33,19,26);(5,8,9,23,11,21,10,22)(6,13,14,32,15,31,24,19)(7,16,17,35,18,34,25,20)(12,26,27,30,29,36,28,33);(2,4)(5,6)(7,12)(8,13)(9,14)(10,24)(11,15)(16,26)(17,27)(18,29)(19,22)(20,33)(21,31)(23,32)(25,28)(30,35)(34,36)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320982 |
| maximal normal subgroup of | #72320982 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333217);