Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497329 |
| 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 | #61060 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#577 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1100 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)(5,24)(6,27,28,11)(7,12)(8,16)(9,22,35,55)(10,25)(14,39)(15,32)(17,45,46,38)(18,30)(20,50)(21,52)(23,49)(26,36,58,57)(29,60,33,63)(31,44,59,43)(34,64,56,54)(37,42,61,51)(40,47)(41,62)(48,53);(1,3,7,18)(2,8,12,32)(4,20,23,14)(5,15,25,16)(6,17,28,46)(9,33,35,29)(10,30,24,13)(11,31,27,59)(19,47,49,62)(21,41,53,40)(22,51,55,42)(26,43,58,44)(34,37,56,61)(36,45,57,38)(39,52,50,48)(54,60,64,63);(1,4,7,23)(2,9,12,35)(3,14,18,20)(5,21,25,53)(6,22,28,55)(8,29,32,33)(10,34,24,56)(11,19,27,49)(13,37,30,61)(15,40,16,41)(17,42,46,51)(26,54,58,64)(31,62,59,47)(36,48,57,52)(38,39,45,50)(43,63,44,60);(3,15)(5,25)(8,30)(10,24)(13,32)(14,40)(16,18)(17,43)(20,41)(21,53)(26,58)(29,61)(31,45)(33,37)(34,56)(36,57)(38,59)(39,47)(42,63)(44,46)(48,52)(50,62)(51,60)(54,64)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,c^b=ci,d^b=de,e^b=ei,g^c=gh,e^d=ei,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312282, #72312289, #72312376, … |
| maximal normal subgroup of | #72312282, #72312289, #72312376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404527);