Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2398 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,16,19,25)(8,13)(9,14)(10,24)(11,15)(12,17,27,20)(18,30,33,35)(21,28)(22,29)(23,34)(26,31,36,32);(1,2,4,3)(5,7)(6,12)(8,18)(9,20)(10,17)(11,19)(13,26)(14,25)(15,27)(16,24)(21,33)(22,32)(23,31)(28,36)(29,35)(30,34);(7,17,19,20)(8,21)(12,25,27,16)(13,28)(18,32,33,31)(22,23)(26,30,36,35)(29,34);(1,3,4,2)(5,8,9,22,11,21,10,23)(6,13,14,29,15,28,24,34)(7,18,20,32,19,33,17,31)(12,26,25,35,27,36,16,30)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 316121);