Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418466 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1861 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)(7,19)(8,14)(9,30)(10,18)(11,16)(12,17)(13,22)(15,33)(20,48)(21,52)(23,38)(24,37)(25,46)(26,56)(27,41)(28,42)(29,40)(31,43)(32,45)(34,44)(35,51)(36,53)(39,60)(47,54)(49,62)(50,63)(55,68)(57,64)(58,66)(59,65)(61,67);(1,2,3,4)(5,7,10,22,12,23,11,24)(6,13,16,19,17,37,18,38)(8,25,27,51,28,20,29,50)(9,31,32,53,34,54,33,21)(14,35,40,46,42,63,41,48)(15,43,44,52,45,47,30,36)(26,57,58,49,59,61,60,55)(39,64,65,68,66,67,56,62);(5,8,12,28)(6,14,17,42)(7,20)(9,26,34,59)(10,27,11,29)(13,35)(15,39,45,66)(16,40,18,41)(19,46)(21,49)(22,50)(23,25)(24,51)(30,56,44,65)(31,61)(32,58,33,60)(36,62)(37,63)(38,48)(43,64)(47,67)(52,68)(53,55)(54,57);(2,4)(5,9,12,34)(6,15,17,45)(7,21,23,53)(8,26,28,59)(10,33,11,32)(13,36,37,52)(14,39,42,66)(16,30,18,44)(19,47,38,43)(20,49,25,55)(22,54,24,31)(27,60,29,58)(35,62,63,68)(40,56,41,65)(46,67,48,64)(50,57,51,61)Polycyclic presentation:
2^9|b^2=e,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330098);