Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503227 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25771 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1100 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)(4,19)(5,9)(6,27)(7,13)(8,14)(10,23)(11,18)(12,29)(16,42)(17,41)(20,47)(21,52)(22,54)(24,49)(25,32)(26,33)(28,59)(30,39)(31,53)(34,51)(35,63)(36,56)(37,45)(38,60)(40,44)(43,48)(46,50)(55,58)(57,62)(61,64);(1,3,6,5,8,18,29,26)(2,9,12,15,14,33,27,11)(4,20,22,50,24,16,43,21)(7,17,28,25,30,45,60,57)(10,31,35,40,36,61,58,34)(13,32,38,41,39,62,59,37)(19,46,48,47,49,52,54,42)(23,44,55,53,56,51,63,64);(1,4,7,23)(2,10,13,19)(3,16,17,44)(5,21,25,53)(6,22,28,55)(8,24,30,56)(9,31,32,52)(11,34,37,47)(12,35,38,48)(14,36,39,49)(15,40,41,42)(18,20,45,51)(26,50,57,64)(27,58,59,54)(29,43,60,63)(33,61,62,46);(1,5,6,18,8,26,29,3)(2,11,12,9,14,15,27,33)(4,21,22,20,24,50,43,16)(7,25,28,45,30,57,60,17)(10,34,35,31,36,40,58,61)(13,37,38,32,39,41,59,62)(19,47,48,52,49,42,54,46)(23,53,55,51,56,64,63,44)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72303765, #72303784, #72303792, … |
| maximal normal subgroup of | #72303765, #72303784, #72303792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410423);