Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16432 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,29,7,13,6,12)(3,14,27,9,18,40,17,33)(4,19,28,36,23,10,22,41)(5,11,25,38,26,39,30,57)(15,42,49,34,20,51,45,31)(16,43,47,60,48,62,56,32)(21,50,24,54,53,35,55,37)(44,63,46,58,52,59,64,61);(1,3)(2,9)(4,20)(5,16)(6,27)(7,18)(8,17)(10,34)(11,32)(12,40)(13,33)(14,29)(15,23)(19,31)(21,52)(22,45)(24,46)(25,56)(26,48)(28,49)(30,47)(35,61)(36,51)(37,59)(38,62)(39,60)(41,42)(43,57)(44,53)(50,58)(54,63)(55,64);(1,4,7,23)(2,10,13,19)(3,15,18,20)(5,24,26,55)(6,28,8,22)(9,31,33,34)(11,37,39,54)(12,41,29,36)(14,42,40,51)(16,46,48,64)(17,49,27,45)(21,30,53,25)(32,59,60,63)(35,57,50,38)(43,58,62,61)(44,56,52,47);(1,5,6,25,7,26,8,30)(2,11,29,57,13,39,12,38)(3,16,27,56,18,48,17,47)(4,21,22,24,23,53,28,55)(9,32,14,43,33,60,40,62)(10,35,41,54,19,50,36,37)(15,44,49,64,20,52,45,46)(31,58,51,59,34,61,42,63)Polycyclic presentation:
2^9|a^2=eg,c^2=i,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369645);