Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C22×#102 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1456 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)(5,6,9,15,10,16,24,30,11,13,25,31,23,32,8,14)(7,17,19,27,20,33,18,34,21,35,22,12,26,28,36,29);(5,7,10,26,11,21,23,20)(6,12,16,34,13,27,32,29)(8,22,9,18,24,19,25,36)(14,33,15,17,30,28,31,35);(2,4);(6,13)(12,27)(14,30)(15,31)(16,32)(17,35)(28,33)(29,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250948, #72250967, #72250969, … |
| maximal normal subgroup of | #72250948, #72250967, #72250969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401632);