Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434424 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15501 |
| 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 | |
1119 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)(3,13)(4,18,22,9)(5,23)(6,27)(7,12)(8,16)(10,26)(11,29)(14,40,19,39)(15,36)(17,38)(20,34,50,44)(21,35,51,45)(24,31)(25,37)(28,32)(30,42,33,47)(41,60,46,62)(43,52)(48,57,54,59)(49,64,63,61)(53,56)(55,58);(1,3)(2,8)(4,19)(5,24)(6,28)(7,17)(9,33)(10,36)(11,38)(12,32)(13,27)(14,22)(15,25)(16,29)(18,30)(20,48)(21,42)(23,52)(26,55)(31,37)(34,60)(35,39)(40,45)(41,63)(43,56)(44,62)(46,49)(47,51)(50,54)(53,58)(57,64)(59,61);(1,4,6,21,7,22,29,51)(2,9,27,45,12,18,11,35)(3,14,16,42,17,19,28,47)(5,20,25,49,26,50,56,63)(8,30,38,40,32,33,13,39)(10,34,53,64,23,44,37,61)(15,41,24,54,43,46,55,48)(31,57,52,60,58,59,36,62);(1,5,6,25,7,26,29,56)(2,10,11,37,12,23,27,53)(3,15,16,24,17,43,28,55)(4,20,21,49,22,50,51,63)(8,31,13,36,32,58,38,52)(9,34,35,61,18,44,45,64)(14,41,42,54,19,46,47,48)(30,57,39,62,33,59,40,60)Polycyclic presentation:
2^9|a^2=g,c^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341620);