Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394044 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3957 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)(6,12)(9,10)(13,17)(14,29)(15,16)(18,32)(19,20)(22,24)(25,30)(26,31)(27,28)(35,36);(2,4)(5,6)(7,17)(8,14)(9,15)(10,12)(11,16)(13,20)(18,31)(19,28)(21,27)(22,25)(23,30)(24,29)(26,33)(32,36)(34,35);(5,7)(6,13)(8,18)(9,20)(10,19)(11,21)(12,17)(14,26)(15,27)(16,28)(22,34)(23,32)(24,33)(25,31)(29,36)(30,35);(5,8)(6,14)(7,18)(9,24)(10,22)(11,23)(12,25)(13,26)(15,29)(16,30)(17,31)(19,34)(20,33)(21,32)(27,36)(28,35)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312259, #72312267, #72312313, … |
| maximal normal subgroup of | #72312259, #72312267, #72312313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301240);