Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,23)(10,14)(11,15)(12,18)(17,32)(19,31)(20,26)(21,29)(22,28)(24,27)(25,33)(30,34)(35,36);(5,7)(6,12,15,26)(8,17)(9,18)(10,20)(11,19)(13,25,28,36)(14,16,23,31)(21,33)(22,34)(24,35)(27,32,29,30);(5,8,10,21,11,22,9,24)(6,13,14,29,15,28,23,27)(7,17,20,33,19,34,18,35)(12,25,16,30,26,36,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72226702, #72226729, #72239644, … |
| maximal normal subgroup of | #72226702, #72226729, #72239644, … |
| maximal quotient of | #72226695, #72249849, C2×#123483 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30679);