Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1132 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)(5,6,9,13,10,14,22,28)(7,15,18,23,17,27,21,11)(8,12,19,25,20,26,33,36)(16,29,31,34,30,35,32,24);(11,23)(15,27)(16,30)(31,32);(5,7,9,21,10,17,22,18)(6,11,13,27,14,23,28,15)(8,16,19,32,20,30,33,31)(12,24,25,35,26,34,36,29);(2,4)(5,8)(6,12)(7,16)(9,19)(10,20)(11,24)(13,25)(14,26)(15,29)(17,30)(18,31)(21,32)(22,33)(23,34)(27,35)(28,36)Polycyclic presentation:
2^9|a^2=f,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72252906, #72252967, #72252977 |
| maximal normal subgroup of | #72252906, #72252967, #72252977 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332203);