Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| 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 | |
857 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)(5,6,12,16,11,18,10,17)(7,19,22,13,24,38,23,39)(8,25,28,43,30,42,29,14)(9,31,35,46,34,45,33,15)(20,36,52,40,48,53,26,55)(21,49,32,44,51,37,50,63)(27,41,58,54,60,66,59,61)(47,64,67,65,57,68,56,62);(5,7,12,22,11,24,10,23)(6,13,16,38,18,39,17,19)(8,26,28,20,30,52,29,48)(9,32,35,51,34,50,33,21)(14,40,25,53,43,55,42,36)(15,44,31,37,46,63,45,49)(27,57,58,56,60,47,59,67)(41,65,54,68,66,62,61,64);(5,8)(6,14,18,43)(7,20)(9,27)(10,28)(11,30)(12,29)(13,36,39,53)(15,41,46,66)(16,42,17,25)(19,40,38,55)(21,47)(22,26)(23,52)(24,48)(31,61,45,54)(32,56)(33,58)(34,60)(35,59)(37,62,49,65)(44,64,63,68)(50,67)(51,57);(2,4)(5,9,12,35,11,34,10,33)(6,15,17,45,18,46,16,31)(7,21,23,50,24,51,22,32)(8,27,29,59,30,60,28,58)(13,37,38,63,39,49,19,44)(14,41,25,54,43,66,42,61)(20,47,52,67,48,57,26,56)(36,62,55,68,53,65,40,64)Polycyclic presentation:
2^9|a^2=efg,b^2=efg,c^2=efi,d^2=ei,e^2=i,f^2=i,g^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,e^b=ei,f^b=fi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349324);