Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455641 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
781 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16)(7,17,22,37)(8,23,27,40)(9,28,33,43)(10,34,30,15)(12,21,38,20)(13,26,41,25)(14,31,44,32)(18,35,50,63)(19,51,55,36)(24,39,59,65)(29,57,58,42)(45,48,56,49)(46,53,60,54)(47,64,67,66)(52,62,61,68);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,21)(11,22)(13,35)(14,36)(15,37)(16,38)(17,34)(20,30)(23,56)(24,47)(25,49)(26,48)(27,50)(28,60)(29,61)(31,54)(32,53)(33,55)(39,62)(40,45)(41,63)(42,66)(43,46)(44,51)(52,58)(57,64)(59,67)(65,68);(2,4)(5,8,10,25,11,27,30,26)(6,13,15,23,16,41,34,40)(7,18,20,48,22,50,21,49)(9,29,31,59,33,58,32,24)(12,35,17,45,38,63,37,56)(14,42,43,65,44,57,28,39)(19,52,53,67,55,61,54,47)(36,64,60,68,51,66,46,62);(5,9,30,31,11,33,10,32)(6,14,34,43,16,44,15,28)(7,19,20,54,22,55,21,53)(8,24,26,29,27,59,25,58)(12,36,17,46,38,51,37,60)(13,39,40,42,41,65,23,57)(18,47,48,61,50,67,49,52)(35,62,45,66,63,68,56,64)Polycyclic presentation:
2^9|a^2=i,b^2=egh,c^2=e,d^2=fi,e^2=i,f^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,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380071);