Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129565 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3058 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1104 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)(7,17,20,32)(8,12)(9,15)(10,26)(11,16)(13,19,31,25)(14,21)(18,30,22,33)(23,28)(24,29)(27,35,36,34);(1,2)(3,4)(6,12)(8,21)(9,10)(13,27)(14,15)(16,29)(17,30)(18,34)(19,25)(22,35)(23,24)(26,28)(31,36)(32,33);(2,4)(5,7)(6,13)(8,22)(9,25)(10,19)(11,20)(12,27)(14,33)(15,32)(16,31)(17,26)(18,24)(21,35)(23,34)(28,30)(29,36)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi,d^c=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249861, #72249872, #72249884, … |
| maximal normal subgroup of | #72249861, #72249872, #72249884, … |
| maximal quotient of | #72265634, C2×#147664 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54860);