Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,12,15)(7,10,16,9)(8,19)(11,26,28,18)(13,29)(14,25,33,24)(17,34)(20,35)(21,31)(22,32)(23,36)(27,30);(1,2,4,3)(5,7,10,15,12,16,9,6)(8,17,22,31,23,30,20,13)(11,26,25,14,28,18,24,33)(19,29,35,27,36,21,32,34);(1,3,4,2)(5,8)(6,13)(7,17)(9,20)(10,22)(11,27)(12,23)(14,32)(15,31)(16,30)(18,19)(21,25)(24,29)(26,36)(28,34)(33,35);(5,9,12,10)(6,7,15,16)(8,20,23,22)(11,24,28,25)(13,17,31,30)(14,18,33,26)(19,35,36,32)(21,27,29,34)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=eh,d^2=i,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250341, #72250355, #72250390, … |
| maximal normal subgroup of | #72250341, #72250355, #72250390, … |
| maximal quotient of | #72319108, #72320047, C2×#494889 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402085);