Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
902 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)(7,15,19,25)(8,12)(9,13)(10,14)(11,18,26,17)(16,31,20,30)(21,27)(22,28)(23,29)(24,33,35,34)(32,36);(5,7,9,17,10,19,23,18)(6,11,13,25,14,26,29,15)(8,20,21,34,22,16,32,33)(12,24,27,31,28,35,36,30);(5,8,9,21,10,22,23,32)(6,12,13,27,14,28,29,36)(7,16,18,34,19,20,17,33)(11,24,15,30,26,35,25,31);(2,4)(7,17,19,18)(11,25,26,15)(16,33,20,34)(24,31,35,30)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322108 |
| maximal normal subgroup of | #72322108 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337427);