Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14171 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,17,9,18,33,29,32)(4,10,22,36,23,38,21,37)(5,24,25,11,27,39,26,40)(15,34,44,55,45,41,19,30)(16,31,48,56,47,57,46,42)(20,50,51,35,52,59,53,60)(43,62,64,61,63,54,49,58);(1,3)(2,9)(4,19)(5,16)(6,29)(7,17)(8,18)(10,34)(11,31)(12,33)(13,32)(14,28)(15,22)(20,49)(21,45)(23,44)(24,42)(25,48)(26,46)(27,47)(30,37)(35,58)(36,55)(38,41)(39,56)(40,57)(43,51)(50,54)(52,64)(53,63)(59,62)(60,61);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,23)(9,30)(11,35)(12,36)(13,38)(14,41)(16,43)(17,44)(18,45)(19,29)(24,50)(25,51)(26,53)(27,52)(28,37)(31,54)(32,55)(33,34)(39,59)(40,60)(42,61)(46,49)(47,63)(48,64)(56,58)(57,62);(1,5,7,25,8,27,6,26)(2,11,12,39,13,40,28,24)(3,16,17,48,18,47,29,46)(4,20,22,51,23,52,21,53)(9,31,33,56,32,57,14,42)(10,35,36,59,38,60,37,50)(15,43,44,64,45,63,19,49)(30,54,34,58,55,62,41,61)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282954);