Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
780 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)(5,6,11,17,12,18,9,15)(7,19,21,14,23,32,22,33)(8,24,25,36,27,13,26,31)(10,29,30,16,20,34,28,35);(5,7,20,8,11,22,28,26,12,23,10,27,9,21,30,25)(6,13,16,32,15,31,29,33,18,24,35,19,17,36,34,14);(2,4)(5,8,10,22,11,26,30,23,12,27,20,21,9,25,28,7)(6,14,16,24,15,32,29,36,18,33,35,13,17,19,34,31);(5,9,12,11)(6,15,18,17)(7,21,23,22)(8,25,27,26)(10,28,20,30)(13,31,24,36)(14,32,33,19)(16,29,35,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=i,e^2=h,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250241, #72250251, #72250310, … |
| maximal normal subgroup of | #72250241, #72250251, #72250310, … |
| maximal quotient of | #72285942, #72309658, #72319279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402124);