Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,19,22,32)(8,14)(9,26,27,36)(10,16)(11,17)(12,18)(13,20,33,21)(15,28,30,23)(24,34)(25,31)(29,35);(5,7)(6,13)(8,23)(9,24)(10,21)(11,20)(12,22)(14,26)(15,35)(16,32)(17,19)(18,33)(25,28)(27,29)(30,34)(31,36);(1,2,4,3)(5,8)(6,14)(7,9)(10,24)(11,29)(12,25)(13,30)(15,33)(16,34)(17,35)(18,31)(19,26)(20,23)(21,28)(22,27)(32,36);(1,3,4,2)(5,9)(6,15)(7,8)(10,28)(11,23)(12,27)(13,31)(14,33)(16,26)(17,36)(18,30)(19,34)(20,29)(21,24)(22,25)(32,35)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280214, #72280644 |
| maximal normal subgroup of | #72280214, #72280644 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292341);