Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1946 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)(5,6,9,14,10,13,21,27)(7,11,17,24,16,23,29,34)(8,12,20,26,19,25,32,35)(15,22,28,33,18,30,31,36);(5,7)(6,11)(8,18,19,15)(9,17)(10,16)(12,22,25,30)(13,23)(14,24)(20,31,32,28)(21,29)(26,33,35,36)(27,34);(1,2,3,4)(5,8)(6,12)(7,15)(9,20)(10,19)(11,22)(13,25)(14,26)(16,18)(17,28)(21,32)(23,30)(24,33)(27,35)(29,31)(34,36);(6,13)(11,23)(12,25)(14,27)(22,30)(24,34)(26,35)(33,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237646, #72237658, #72237664, … |
| maximal normal subgroup of | #72237646, #72237658, #72237664, … |
| maximal quotient of | #72314605 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265653);