Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497323 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61052 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#577 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1109 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)(4,19)(5,23)(6,26,27,11)(7,12)(8,16)(9,21,33,52)(10,24)(14,37)(15,31)(17,44,45,36)(18,29)(20,49)(22,48)(25,34,55,54)(28,57,59,63)(30,43,56,42)(32,60,53,51)(35,40,58,64)(38,61)(39,46)(41,62)(47,50);(1,3,7,18)(2,8,12,31)(4,14,22,41)(5,15,24,16)(6,17,27,45)(9,28,33,59)(10,29,23,13)(11,30,26,56)(19,46,48,61)(20,38,50,39)(21,40,52,64)(25,42,55,43)(32,58,53,35)(34,44,54,36)(37,47,62,49)(51,63,60,57);(1,4,7,22)(2,9,12,33)(3,14,18,41)(5,20,24,50)(6,21,27,52)(8,28,31,59)(10,32,23,53)(11,19,26,48)(13,35,29,58)(15,38,16,39)(17,40,45,64)(25,51,55,60)(30,46,56,61)(34,47,54,49)(36,37,44,62)(42,63,43,57);(3,15)(5,24)(8,29)(10,23)(13,31)(14,38)(16,18)(17,42)(20,50)(25,55)(28,58)(30,44)(32,53)(34,54)(35,59)(36,56)(37,61)(39,41)(40,63)(43,45)(46,62)(47,49)(51,60)(57,64)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,g^c=gh,e^d=ei,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72312286, #72312289, #72312379, … |
| maximal normal subgroup of | #72312286, #72312289, #72312379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404522);