Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1547 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,18)(8,23)(9,16)(10,15)(11,17)(12,20)(13,25)(14,22)(19,29)(21,28)(24,32)(26,31)(27,36)(30,34)(33,35);(5,7,9,20,11,22,10,21)(6,12,15,18,17,28,16,14)(8,19,24,34,26,36,25,35)(13,29,31,33,32,27,23,30);(1,2,3,4)(5,8)(6,13)(7,19)(9,25)(10,24)(11,26)(12,27)(14,30)(15,23)(16,31)(17,32)(18,33)(20,35)(21,34)(22,36)(28,29);(5,7)(6,14)(8,19)(9,21)(10,20)(11,22)(12,16)(13,30)(15,28)(17,18)(23,29)(24,35)(25,34)(26,36)(27,31)(32,33)Polycyclic presentation:
2^9|b^2=e,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318217, #72318867, #72318957 |
| maximal normal subgroup of | #72318217, #72318867, #72318957 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330416);