Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,13)(7,11,16,26)(8,18)(9,14,23,29)(12,20)(15,24)(17,25,32,36)(19,27)(21,30)(22,28)(31,35)(33,34);(2,4)(5,7,10,16)(6,11,13,26)(8,19)(9,17,23,32)(12,27)(14,25,29,36)(15,22)(18,34)(20,33)(21,31)(24,30)(28,35);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,18)(14,30)(16,31)(17,33)(19,32)(21,23)(25,34)(26,35)(27,36)(28,29);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,26)(14,25)(18,35)(19,21)(20,31)(22,33)(23,32)(27,28)(29,36)(30,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^b=fi,e^c=ei,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239333, #72265644, #72265655, … |
| maximal normal subgroup of | #72239333, #72265644, #72265655, … |
| maximal quotient of | #72279929, #72280013, #72280051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271738);