Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| 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 | |
897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,18)(7,19,24,38)(8,25,30,42)(9,31,35,45)(10,17,11,16)(13,23,39,22)(14,28,43,29)(15,34,46,33)(20,36,51,56)(21,37,55,65)(26,47,50,40)(27,41,59,67)(32,60,58,44)(48,54,64,53)(49,66,61,63)(52,68,57,62);(1,2,3,4)(5,7)(6,13)(8,26)(9,21)(10,23)(11,22)(12,24)(14,40)(15,37)(16,38)(17,19)(18,39)(20,28)(25,56)(27,57)(29,51)(30,50)(31,48)(32,61)(33,54)(34,53)(35,55)(36,42)(41,66)(43,47)(44,68)(45,64)(46,65)(49,58)(52,59)(60,62)(63,67);(2,4)(5,8)(6,14)(7,20,24,51)(9,32)(10,29)(11,28)(12,30)(13,36,39,56)(15,44)(16,25)(17,42)(18,43)(19,47,38,40)(21,52,55,57)(22,50,23,26)(27,33)(31,41)(34,59)(35,58)(37,63,65,66)(45,67)(46,60)(48,68,64,62)(49,54,61,53);(5,9,12,35)(6,15,18,46)(7,21,24,55)(8,27,30,59)(10,34,11,33)(13,37,39,65)(14,41,43,67)(16,45,17,31)(19,48,38,64)(20,49,51,61)(22,54,23,53)(25,44,42,60)(26,57,50,52)(28,58,29,32)(36,62,56,68)(40,66,47,63)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379480);