Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16357 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1157 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)(4,10)(5,26,29,40)(6,13)(7,12)(8,14)(9,18)(11,28,41,30)(16,43)(17,32,49,61)(19,34)(20,33)(21,31)(22,53,54,39)(23,42)(24,37)(25,38)(27,56,55,36)(35,46)(44,50,60,48)(45,57,64,62)(47,63,52,59)(51,58);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,21,23,46,25,51,24,16)(5,17,30,50,29,49,28,48)(10,35,37,31,38,43,42,58)(11,32,26,44,41,61,40,60)(22,52,55,64,54,47,27,45)(36,59,53,57,56,63,39,62);(1,4)(2,10)(3,16)(5,27,29,55)(6,23)(7,24)(8,25)(9,31)(11,39,41,53)(12,37)(13,42)(14,38)(15,43)(17,47,49,52)(18,21)(19,51)(20,46)(22,28,54,30)(26,36,40,56)(32,59,61,63)(33,35)(34,58)(44,57,60,62)(45,48,64,50);(1,5,6,30,8,29,7,28)(2,11,12,26,14,41,13,40)(3,17,18,50,20,49,19,48)(4,22,23,55,25,54,24,27)(9,32,15,44,34,61,33,60)(10,36,37,53,38,56,42,39)(16,45,21,52,46,64,51,47)(31,57,43,63,58,62,35,59)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72284248, #72319687, #72320000, … |
| maximal normal subgroup of | #72284248, #72319687, #72320000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367323);