Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,25)(7,12,19,28,17,27,18,34)(8,20,23,30,24,36,22,13)(16,31,33,29,21,35,32,26);(1,2)(3,4)(5,7)(6,12)(8,21)(9,18)(10,19)(11,17)(13,29)(14,28)(15,27)(16,24)(20,35)(22,33)(23,32)(25,34)(26,30)(31,36);(5,8,9,22,11,24,10,23)(6,13,14,20,15,30,25,36)(7,16,18,32,17,21,19,33)(12,26,28,31,27,29,34,35);(2,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,26)(14,28)(15,27)(20,31)(21,24)(22,32)(23,33)(25,34)(29,30)(35,36);(7,17)(12,27)(16,21)(18,19)(26,29)(28,34)(31,35)(32,33)Polycyclic presentation:
2^9|a^2=h,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72343550, #72343653, #72343656 |
| maximal normal subgroup of | #72343550, #72343653, #72343656 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287299);