Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,11)(8,20,21,12)(9,14)(10,13)(16,23,27,19)(17,26)(22,33,34,28)(24,25)(29,32)(30,35)(31,36);(5,7)(6,11)(8,17)(9,23)(10,18)(12,26)(13,15)(14,16)(19,25)(20,29)(21,32)(22,35)(24,27)(28,30)(31,34)(33,36);(2,4)(5,6)(7,16)(8,12)(9,24)(10,13)(11,23)(14,25)(15,19)(17,30)(18,27)(20,21)(22,33)(26,35)(28,34)(29,31)(32,36);(5,8,10,21)(6,12,13,20)(7,17,18,32)(9,22,25,34)(11,26,15,29)(14,28,24,33)(16,30,27,36)(19,31,23,35)Polycyclic presentation:
2^9|a^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,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 | #72313435 |
| maximal normal subgroup of | #72313435 |
| maximal quotient of | #72313446, #72313449, #72313493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266743);