Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1752 |
| prime radical | C24 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,8,9,14,24,22,12,15,11,23,26,28,25,36)(7,16,19,33,20,13,29,30,21,31,27,34,18,32,35,17);(1,2,3,4)(5,7)(6,13)(8,17)(9,20)(10,27)(11,19)(12,21)(14,31)(15,32)(16,28)(18,26)(22,33)(23,30)(24,35)(25,29)(34,36);(5,8,11,14,9,22,25,15,12,23,10,28,26,36,24,6)(7,17,19,31,20,33,29,32,21,30,27,16,18,34,35,13)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ch,g^a=gi,d^b=di,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72314229, #72314281 |
| maximal normal subgroup of | #72314229, #72314281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53522);