Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1655 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)(8,21)(9,27)(10,28)(11,15)(12,16)(13,19)(14,25)(18,35)(20,30)(22,34)(23,31)(24,33)(26,32)(29,36);(5,7)(6,13)(8,22)(9,10)(11,19)(12,20)(14,31)(15,30)(16,27)(17,28)(18,24)(21,29)(23,26)(25,36)(32,35)(33,34);(2,4)(5,8)(6,14)(7,18)(9,23)(10,24)(11,25)(12,26)(13,29)(15,32)(16,33)(17,34)(19,22)(20,36)(21,28)(27,35)(30,31);(5,9,11,20,12,19,10,7)(6,13,28,30,16,27,15,17)(8,23,25,36,26,22,24,18)(14,29,21,31,33,35,32,34)Polycyclic presentation:
2^9|d^2=ei,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,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325705);