Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481349 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1294 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)(6,11,14,19)(8,12)(9,21)(13,22,27,32)(15,24,29,23)(17,25)(18,31)(20,26)(28,34,36,33)(30,35);(5,6,12,24,21,29,8,19,10,14,26,23,9,15,20,11)(7,13,25,34,31,36,17,32,16,27,35,33,18,28,30,22);(1,2,3,4)(5,7)(6,13)(8,17)(9,18)(10,16)(11,22)(12,25)(14,27)(15,28)(19,32)(20,30)(21,31)(23,33)(24,34)(26,35)(29,36);(7,16)(13,27)(17,30)(18,31)(22,32)(25,35)(28,36)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224826, #72226813, #72226819, … |
| maximal normal subgroup of | #72224826, #72226813, #72226819, … |
| maximal quotient of | #72224821, #72310125, #72319081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402331);