Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,10,16,11,15,30,40)(7,12,19,34,20,33,49,60)(8,21,24,41,25,13,29,37)(9,26,27,55,28,14,38,39)(17,42,22,50,44,31,48,35)(18,45,46,66,47,32,58,59)(23,36,53,62,54,51,56,63)(43,57,52,61,65,64,67,68);(1,2)(3,4)(5,7,10,19,11,20,30,49)(6,12,16,34,15,33,40,60)(8,22,24,44,25,48,29,17)(9,18,27,46,28,47,38,58)(13,35,37,42,21,50,41,31)(14,32,39,59,26,45,55,66)(23,52,53,65,54,67,56,43)(36,61,62,64,51,68,63,57);(5,8,11,25)(6,13,15,21)(7,17,20,44)(9,23,28,54)(10,29,30,24)(12,31,33,42)(14,36,26,51)(16,41,40,37)(18,43,47,65)(19,48,49,22)(27,56,38,53)(32,57,45,64)(34,50,60,35)(39,63,55,62)(46,67,58,52)(59,68,66,61);(2,4)(5,9)(6,14,15,26)(7,18)(8,23)(10,27)(11,28)(12,32,33,45)(13,36,21,51)(16,39,40,55)(17,43)(19,46)(20,47)(22,52)(24,53)(25,54)(29,56)(30,38)(31,57,42,64)(34,59,60,66)(35,61,50,68)(37,62,41,63)(44,65)(48,67)(49,58)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349348);