Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454039 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2198 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)(5,6,11,16)(7,17,20,28)(8,22,24,29)(9,15,10,14)(12,21,27,19)(13,23,30,25)(18,32,34,26)(31,33,36,35);(5,7)(6,12)(8,18)(9,21)(10,19)(11,20)(13,26)(14,28)(15,17)(16,27)(22,36)(23,35)(24,34)(25,33)(29,31)(30,32);(8,23)(9,10)(13,29)(14,15)(17,28)(18,33)(19,21)(22,30)(24,25)(26,36)(31,32)(34,35);(2,4)(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,26)(14,22)(15,29)(16,30)(17,31)(19,35)(20,34)(21,33)(27,32)(28,36)Polycyclic presentation:
2^9|a^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318890, #72318894, #72318957 |
| maximal normal subgroup of | #72318890, #72318894, #72318957 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361287);