Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394463 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2257 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)(7,16)(8,21)(9,26)(10,14)(11,15)(12,18)(13,23)(17,27)(19,29)(20,28)(22,30)(24,31)(25,34)(32,35)(33,36);(5,7)(6,12)(8,22)(9,19)(10,18)(11,20)(13,30)(14,28)(15,29)(16,26)(17,24)(21,27)(23,33)(25,32)(31,35)(34,36);(5,8,9,23,11,25,10,24)(6,13,14,21,15,31,26,34)(7,17,18,32,20,33,19,22)(12,27,16,30,29,36,28,35);(2,4)Polycyclic presentation:
2^9|c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252890, #72256881, #72256910 |
| maximal normal subgroup of | #72252890, #72256881, #72256910 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301802);