Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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×#3027 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15,19,11)(8,20)(9,24)(10,13)(12,23)(14,26)(16,30,34,36)(17,28,18,29)(21,27,35,33)(22,32)(25,31);(1,2,3,4)(5,7)(6,11)(8,21)(9,17)(10,19)(12,30)(13,15)(14,28)(16,23)(18,26)(20,27)(22,35)(24,29)(25,34)(31,36)(32,33);(2,4)(5,8)(6,12)(7,16)(9,25)(10,22)(11,27)(13,31)(14,32)(15,33)(17,21)(18,35)(19,34)(20,24)(23,26)(28,30)(29,36);(5,7)(6,11)(8,16)(9,18)(10,19)(12,27)(13,15)(14,29)(17,26)(20,30)(21,23)(22,34)(24,28)(25,35)(31,33)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280941, #72281002, #72281366, … |
| maximal normal subgroup of | #72280941, #72281002, #72281366, … |
| maximal quotient of | #72313370 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273840);