Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1794 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)(8,18)(9,21)(10,14)(12,20)(13,23)(15,30)(16,33)(17,26)(19,29)(22,28)(24,32)(25,35)(27,34)(31,36);(5,7,10,17)(6,11,14,26)(8,15,19,31)(9,16,23,35)(12,27,28,24)(13,25,21,33)(18,36,29,30)(20,32,22,34);(5,8)(6,12)(7,15)(9,22)(10,19)(11,24)(13,29)(14,28)(16,34)(17,31)(18,21)(20,23)(25,36)(26,27)(30,33)(32,35);(1,3,4,2)(9,23)(13,21)(16,35)(18,29)(20,22)(25,33)(30,36)(32,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312515, #72312596 |
| maximal normal subgroup of | #72312515, #72312596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282975);