Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4032 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)(7,17)(8,22)(9,15)(10,14)(11,16)(12,20)(13,23)(18,27)(19,28)(21,29)(24,31)(25,30)(26,32)(33,36)(34,35);(2,4)(5,7)(6,12)(8,19)(9,25)(10,20)(11,21)(13,28)(14,29)(15,17)(16,30)(18,23)(22,35)(24,33)(26,34)(27,31)(32,36);(5,8)(6,13)(7,18)(9,26)(10,23)(11,24)(12,27)(14,31)(15,22)(16,32)(17,28)(19,25)(20,33)(21,34)(29,36)(30,35);(5,8)(6,13)(7,19)(9,23)(10,26)(11,24)(12,28)(14,22)(15,31)(16,32)(17,27)(18,25)(20,34)(21,33)(29,35)(30,36)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72255594, #72256284, #72256291, … |
| maximal normal subgroup of | #72255594, #72256284, #72256291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301251);