Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1974 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)(5,6,9,14,10,13,20,26)(7,11,17,23,16,21,29,32)(8,12,18,25,19,24,30,34)(15,22,28,33,27,31,35,36);(11,21)(22,31)(23,32)(33,36);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,22)(13,21)(14,23)(18,28)(19,27)(20,29)(24,31)(25,33)(26,32)(30,35)(34,36);(5,8,9,18,10,19,20,30)(6,12,14,25,13,24,26,34)(7,15,17,28,16,27,29,35)(11,22,23,33,21,31,32,36)Polycyclic presentation:
2^9|a^2=h,d^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72231526, #72249848, #72251058, … |
| maximal normal subgroup of | #72231526, #72249848, #72251058, … |
| maximal quotient of | C2×#359028, #72314526, #72343059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266224);