Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60415 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,15,11,14,9,23)(7,16,18,12,19,27,24,26)(8,13,21,29,22,28,20,33)(17,30,31,25,32,36,34,35);(1,2,3,4);(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,25)(14,26)(15,27)(16,23)(20,34)(21,31)(22,32)(28,35)(29,36)(30,33);(5,8,10,21,11,22,9,20)(6,13,15,29,14,28,23,33)(7,17,18,31,19,32,24,34)(12,25,27,36,26,35,16,30);(6,14)(12,26)(13,28)(15,23)(16,27)(25,35)(29,33)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72285370, #72285487, D4×#2955, … |
| maximal normal subgroup of | #72285370, #72285487, D4×#2955, … |
| maximal quotient of | #72285487, #72285591, #72285957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280033);