Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503229 |
| 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 | |
1103 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)(3,11)(4,10)(5,24)(6,27)(7,29)(8,14)(9,16)(12,25)(13,31)(15,35)(17,47)(18,40)(19,49)(20,52)(21,55)(22,37)(23,38)(26,33)(28,41)(30,39)(32,44)(34,50)(36,53)(42,46)(43,62)(45,61)(48,57)(51,64)(54,60)(56,58)(59,63);(1,3,6,16,8,18,25,5)(2,9,12,11,14,24,27,40)(4,19,21,50,23,51,53,54)(7,30,28,26,31,17,48,46)(10,34,36,49,38,60,55,64)(13,42,41,47,29,33,57,39)(15,43,44,56,45,63,20,22)(32,59,35,58,52,62,61,37);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22,31,56)(8,23)(9,32)(11,35)(12,36)(13,37,29,58)(14,38)(16,44)(17,19,30,51)(18,45)(24,52)(25,53)(26,54,46,50)(27,55)(28,43,48,63)(33,34,42,60)(39,49,47,64)(40,61)(41,59,57,62);(1,5,25,18,8,16,6,3)(2,11,27,9,14,40,12,24)(4,20,53,45,23,44,21,15)(7,26,48,30,31,46,28,17)(10,35,55,32,38,61,36,52)(13,39,57,33,29,47,41,42)(19,22,54,63,51,56,50,43)(34,58,64,59,60,37,49,62)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312282, #72312316, #72312376, … |
| maximal normal subgroup of | #72312282, #72312316, #72312376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410427);