Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1960 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)(5,6,10,14,11,15,9,23)(7,16,18,12,19,27,24,25)(8,13,21,28,22,29,20,33)(17,30,31,26,32,36,34,35);(2,4)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,26)(14,27)(15,25)(16,23)(20,34)(21,31)(22,32)(28,36)(29,35)(30,33);(12,25)(16,27)(26,35)(30,36);(5,8,10,21,11,22,9,20)(6,13,14,28,15,29,23,33)(7,17,18,31,19,32,24,34)(12,26,27,36,25,35,16,30)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72231526, #72249848, #72250083, … |
| maximal normal subgroup of | #72231526, #72249848, #72250083, … |
| maximal quotient of | C2×#359014, #72308380, #72343218 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266210);