Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1838 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)(7,11)(8,18,21,12)(9,22)(10,13)(14,25)(15,26,33,36)(16,27)(17,31)(19,35,34,29)(20,30,24,32)(23,28);(1,2,4,3)(5,7)(6,11)(8,19)(9,23)(10,16)(12,29)(13,27)(14,31)(15,20)(17,25)(18,35)(21,34)(22,28)(24,33)(26,30)(32,36);(1,3,4,2)(5,8)(6,12)(7,15)(9,24)(10,21)(11,26)(13,18)(14,32)(16,33)(17,34)(19,23)(20,25)(22,30)(27,36)(28,35)(29,31);(7,16)(11,27)(15,33)(17,23)(19,34)(26,36)(28,31)(29,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316895, #72316901, #72316909, … |
| maximal normal subgroup of | #72316895, #72316901, #72316909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265270);