Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12750 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,17,20,35)(8,22)(9,13)(10,27)(11,16)(12,18,31,25)(14,24)(15,26)(19,34,36,32)(21,29,28,30)(23,33);(1,2,3,4)(5,7)(6,12)(8,19)(9,25)(10,28)(11,20)(13,29)(14,17)(15,34)(16,31)(18,23)(21,26)(22,35)(24,36)(27,32)(30,33);(6,13)(8,23)(9,24)(10,26)(12,29)(14,15)(16,33)(17,34)(18,19)(21,28)(22,27)(25,36)(30,31)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249880, #72249887, #72249913, … |
| maximal normal subgroup of | #72249880, #72249887, #72249913, … |
| maximal quotient of | #72265658, #72280015, C2×#146767 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53963);