Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136643 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,20,35)(8,22)(9,14)(10,27)(11,15)(12,18,29,36)(13,25)(17,33,28,23)(19,34,21,30)(24,31)(26,32);(5,7)(6,12)(8,23)(9,21)(10,19)(11,20)(13,30)(14,17)(15,29)(16,32)(18,24)(22,35)(25,36)(26,33)(27,28)(31,34);(2,4)(5,6)(7,17)(8,13)(9,27)(10,14)(11,15)(12,21)(16,33)(18,30)(19,29)(20,28)(22,25)(23,35)(24,32)(26,31)(34,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dhi,f^b=fi,g^b=gi,h^b=hi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249860, #72249861, #72249885, … |
| maximal normal subgroup of | #72249860, #72249861, #72249885, … |
| maximal quotient of | #72265025, C2×#147673 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54869);