Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| 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 | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,10,16,12,18,11,17)(7,13,21,37,23,39,22,38)(8,24,27,43,29,42,28,14)(9,30,33,45,32,46,34,15)(19,40,25,55,49,64,48,35)(20,50,53,36,52,60,54,66)(26,41,58,61,31,44,59,56)(47,65,62,67,51,63,57,68);(1,2,3,4)(5,7)(6,13)(8,25)(9,20)(10,21)(11,22)(12,23)(14,40)(15,36)(16,37)(17,38)(18,39)(19,28)(24,55)(26,57)(27,49)(29,48)(30,60)(31,62)(32,52)(33,53)(34,54)(35,42)(41,67)(43,64)(44,68)(45,66)(46,50)(47,58)(51,59)(56,65)(61,63);(2,4)(5,8,12,29)(6,14,18,43)(7,19,23,49)(9,31,32,26)(10,28,11,27)(13,35,39,55)(15,44,45,41)(16,42,17,24)(20,51,52,47)(21,48,22,25)(30,61,46,56)(33,58,34,59)(36,65,66,63)(37,64,38,40)(50,67,60,68)(53,62,54,57);(5,9,10,34,12,32,11,33)(6,15,16,46,18,45,17,30)(7,20,21,54,23,52,22,53)(8,26,27,59,29,31,28,58)(13,36,37,50,39,66,38,60)(14,41,24,56,43,44,42,61)(19,47,25,57,49,51,48,62)(35,63,40,67,55,65,64,68)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380331);