Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213778 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2476 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)(7,15,19,12)(8,20)(9,23,25,13)(10,22)(11,14)(16,31,30,18)(17,27)(21,24,28,26)(29,34)(32,33)(35,36);(2,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,19)(13,27)(14,15)(20,31)(21,32)(22,30)(23,29)(24,33)(25,34)(26,35)(28,36);(5,8,20,6,11,22,10,14)(7,16,18,15,19,30,31,12)(9,21,24,13,25,28,26,23)(17,32,35,29,34,36,33,27);(2,4)(5,9,11,25)(6,13,14,23)(7,17,19,34)(8,21,22,28)(10,26,20,24)(12,27,15,29)(16,32,30,36)(18,35,31,33)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318058, #72318110, #72318131, … |
| maximal normal subgroup of | #72318058, #72318110, #72318131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281231);