Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33516 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1054 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)(5,6,10,14,11,16,27,15)(7,17,20,32,22,36,21,29)(8,13,23,33,24,34,9,25)(12,28,30,18,31,26,35,19);(5,7)(6,12)(8,19)(9,26)(10,20)(11,22)(13,32)(14,30)(15,35)(16,31)(17,25)(18,24)(21,27)(23,28)(29,34)(33,36);(2,4);(5,8,10,23,11,24,27,9)(6,13,14,33,16,34,15,25)(7,18,20,26,22,19,21,28)(12,29,30,17,31,32,35,36)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=eh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72287284, D4×#2179, #72319558, … |
| maximal normal subgroup of | #72287284, D4×#2179, #72319558, … |
| maximal quotient of | #72286666, #72321045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402891);