Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,16,11,18,10,17)(7,19,22,13,24,38,23,37)(8,14,29,41,28,43,27,42)(9,30,34,46,33,45,32,15)(20,48,25,35,51,39,50,61)(21,36,53,47,55,64,54,63)(26,44,58,66,57,65,31,40)(49,60,52,68,59,67,56,62);(1,2,3,4)(5,7)(6,13)(8,25)(9,21)(10,23)(11,24)(12,22)(14,39)(15,36)(16,38)(17,19)(18,37)(20,27)(26,56)(28,50)(29,51)(30,47)(31,59)(32,54)(33,55)(34,53)(35,42)(40,62)(41,61)(43,48)(44,60)(45,63)(46,64)(49,58)(52,57)(65,67)(66,68);(2,4)(5,8,11,28)(6,14,18,43)(7,20,24,51)(9,31,33,58)(10,29,12,27)(13,35,37,61)(15,44,46,65)(16,42,17,41)(19,39,38,48)(21,52,55,56)(22,50,23,25)(26,34,57,32)(30,40,45,66)(36,62,64,68)(47,67,63,60)(49,54,59,53);(5,9,11,33)(6,15,18,46)(7,21,24,55)(8,26,28,57)(10,34,12,32)(13,36,37,64)(14,40,43,66)(16,45,17,30)(19,47,38,63)(20,49,51,59)(22,54,23,53)(25,56,50,52)(27,58,29,31)(35,60,61,67)(39,62,48,68)(41,65,42,44)Polycyclic presentation:
2^9|a^2=ef,b^2=gh,c^2=i,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378982);