Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,20)(9,25)(10,15)(12,23)(13,18)(14,22)(16,29)(17,28)(19,27)(21,33)(24,35)(26,31)(30,36)(32,34);(2,3)(5,7)(6,11)(8,21)(9,18)(10,19)(12,26)(13,25)(14,28)(15,27)(16,24)(17,22)(20,33)(23,31)(29,35)(30,34)(32,36);(1,2,4,3)(5,8,22,36,10,24,9,23)(6,12,25,35,15,30,14,20)(7,16,18,32,19,21,17,31)(11,26,28,33,27,34,13,29);(1,3,4,2)(5,7)(6,13)(8,16)(9,17)(10,19)(11,25)(12,29)(14,27)(15,28)(18,22)(20,34)(21,24)(23,31)(26,35)(30,33)(32,36)Polycyclic presentation:
2^9|a^2=e,c^2=efi,d^2=e,f^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=de,f^b=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331456);