Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
822 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)(7,15,19,25)(8,12)(9,13)(10,14)(11,18,26,17)(16,31,20,30)(21,27)(22,28)(23,29)(24,32,35,33)(34,36);(5,7)(6,11)(8,20)(9,17)(10,19)(12,24)(13,25)(14,26)(15,29)(16,22)(18,23)(21,33)(27,31)(28,35)(30,36)(32,34);(5,8,9,21,10,22,23,34)(6,12,13,27,14,28,29,36)(7,16,18,33,19,20,17,32)(11,24,15,30,26,35,25,31);(2,4)(7,17,19,18)(11,25,26,15)(16,32,20,33)(24,31,35,30)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320778 |
| maximal normal subgroup of | #72320778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335798);