Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427391 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15369 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1205 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)(3,8)(4,17,20,9)(5,10,23,37)(6,12)(7,11)(13,29,44,56)(14,36,22,30)(15,32)(16,31)(18,33,46,48)(19,49)(21,41,26,35)(24,38,27,42)(25,40)(28,39)(34,52)(43,62)(45,58,47,57)(50,64)(51,61)(53,60)(54,55)(59,63);(1,3,7,15)(2,8,11,32)(4,18,20,46)(5,22,23,14)(6,25,28,16)(9,33,17,48)(10,36,37,30)(12,40,39,31)(13,26,44,21)(19,50,52,63)(24,45,27,47)(29,41,56,35)(34,59,49,64)(38,57,42,58)(43,51,54,53)(55,60,62,61);(1,4)(2,9)(3,13)(5,19)(6,26)(7,20)(8,29)(10,34)(11,17)(12,41)(14,43)(15,44)(16,46)(18,25)(21,28)(22,54)(23,52)(24,53)(27,51)(30,55)(31,48)(32,56)(33,40)(35,39)(36,62)(37,49)(38,61)(42,60)(45,50)(47,63)(57,59)(58,64);(1,5)(2,10,11,37)(3,14,15,22)(4,19)(6,27)(7,23)(8,30)(9,34,17,49)(12,42,39,38)(13,43,44,54)(16,47,25,45)(18,50,46,63)(20,52)(21,53)(24,28)(26,51)(29,55)(31,58)(32,36)(33,59)(35,61,41,60)(40,57)(48,64)(56,62)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72286066, #72286106, #72286154, … |
| maximal normal subgroup of | #72286066, #72286106, #72286154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334588);