Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
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 | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4398 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)(7,15,18,11)(8,12)(9,22)(10,13)(14,25)(16,33,34,26)(17,27,23,31)(19,28,35,36)(20,29)(21,32)(24,30);(5,7)(6,11)(8,19)(9,23)(10,18)(12,28)(13,15)(14,31)(16,21)(17,25)(20,35)(22,27)(24,34)(26,30)(29,36)(32,33);(2,4)(5,8)(6,12)(7,16)(9,24)(10,20)(11,26)(13,29)(14,32)(15,33)(17,35)(18,34)(19,23)(21,25)(22,30)(27,36)(28,31);(8,20)(12,29)(16,34)(19,35)(21,24)(26,33)(28,36)(30,32)Polycyclic presentation:
2^9|a^2=h,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237503, #72237510, #72237695, … |
| maximal normal subgroup of | #72237503, #72237510, #72237695, … |
| maximal quotient of | #72239460, #72279767, #72281497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264773);