Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #13742 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| prime radical | C22×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2,3,4)(5,6)(7,14)(8,13)(9,27)(10,31)(11,33)(12,18)(15,22)(16,23)(17,28)(19,34)(20,50)(21,45)(24,44)(25,38)(26,37)(29,60)(30,59)(32,61)(35,53)(36,52)(39,48)(40,47)(41,56)(42,54)(43,55)(46,62)(49,65)(51,58)(57,66)(63,67)(64,68);(6,13)(7,11)(8,22)(9,23)(10,28)(14,38)(15,16)(17,44)(18,37)(19,46)(20,51)(21,32)(24,56)(25,48)(26,54)(27,35)(29,47)(30,55)(31,36)(33,34)(39,58)(40,62)(41,50)(42,43)(45,66)(49,68)(52,53)(57,67)(59,64)(60,63)(61,65);(5,7,12,11)(6,14,18,33)(8,19,26,25)(9,20,30,29)(10,21,32,28)(13,34,37,38)(15,39,42,62)(16,40,43,58)(17,31,45,61)(22,48,54,46)(23,47,55,51)(24,49,57,52)(27,50,59,60)(35,63,64,41)(36,44,65,66)(53,67,68,56)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=df,f^b=fg,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286491, #72286843, #72286862, … |
| maximal normal subgroup of | #72286491, #72286843, #72286862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60753);