Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6)(7,19,23,38)(8,24)(9,15)(10,16)(11,17)(12,18)(13,22,39,35)(14,27)(20,49,51,36)(21,52,30,48)(25,40,55,62)(26,56)(28,43)(29,42)(31,41)(32,45)(33,46)(34,47)(37,60,64,54)(44,59)(50,61,68,67)(53,63,57,65)(58,66);(5,7,12,23)(6,13)(8,25,29,55)(9,30,34,21)(10,22,11,35)(14,40)(15,37)(16,19)(17,38)(18,39)(20,27,51,28)(24,49)(26,57,59,53)(31,50,58,68)(32,60,33,54)(36,42)(41,65)(43,62)(44,61)(45,48)(46,52)(47,64)(56,67)(63,66);(5,8,12,29)(6,14,18,43)(7,20,23,51)(9,31,34,58)(10,28,11,27)(13,36,39,49)(15,44,47,56)(16,24,17,42)(19,40,38,62)(21,53,30,57)(22,55,35,25)(26,33,59,32)(37,63,64,65)(41,46,66,45)(48,61,52,67)(50,60,68,54);(2,4)(5,9,11,33,12,34,10,32)(6,15,17,46,18,47,16,45)(7,21,35,60,23,30,22,54)(8,26,28,58,29,59,27,31)(13,37,38,52,39,64,19,48)(14,41,24,56,43,66,42,44)(20,50,55,57,51,68,25,53)(36,61,40,65,49,67,62,63)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385674);