Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13396 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#2034 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1061 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,21)(5,11)(6,27)(7,12)(8,14)(9,17)(10,24)(13,29)(16,45)(18,34)(19,33)(20,35)(22,56)(23,41)(25,55)(26,42)(28,38)(30,40)(31,54)(32,61)(36,49)(37,58)(39,59)(43,47)(44,51)(46,63)(48,60)(50,64)(52,53)(57,62);(1,3,7,20,8,19,5,17)(2,9,11,33,14,35,12,15)(4,22,23,52,25,16,31,46)(6,18,29,51,30,50,26,48)(10,36,38,62,39,32,37,43)(13,34,27,60,42,64,40,44)(21,53,54,56,55,63,41,45)(24,57,58,49,59,47,28,61);(1,4,5,23,8,25,7,31)(2,10,12,38,14,39,11,37)(3,16,17,46,19,22,20,52)(6,28,26,24,30,58,29,59)(9,32,15,43,35,36,33,62)(13,41,40,21,42,54,27,55)(18,49,48,47,50,61,51,57)(34,56,44,63,64,45,60,53);(1,5,8,7)(2,11,14,12)(3,17,19,20)(4,23,25,31)(6,26,30,29)(9,33,35,15)(10,37,39,38)(13,27,42,40)(16,46,22,52)(18,48,50,51)(21,41,55,54)(24,58,59,28)(32,62,36,43)(34,60,64,44)(45,63,56,53)(47,61,57,49)Polycyclic presentation:
2^9|b^2=h,c^2=e,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=cg,g^b=gi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249772, #72249781, #72249836, … |
| maximal normal subgroup of | #72249772, #72249781, #72249836, … |
| maximal quotient of | #72250211, C2×#499050, #72322928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406246);