Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499885 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1825 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)(3,14)(4,9)(5,10)(6,27)(7,13)(8,16)(11,25)(12,30)(15,46)(17,40)(18,51)(19,53)(20,37)(21,38)(22,43)(23,33)(24,35)(26,41)(28,45)(29,36)(31,49)(32,57)(34,54)(39,59)(42,50)(44,47)(48,60)(52,62)(55,61)(56,63)(58,64);(1,3)(2,8)(4,19)(5,23)(6,28)(7,17)(9,34)(10,14)(11,40)(12,42)(13,33)(15,22)(16,25)(18,30)(20,52)(21,55)(24,54)(26,50)(27,32)(29,47)(31,36)(35,62)(37,61)(38,53)(39,51)(41,45)(43,60)(44,64)(46,63)(48,58)(49,56)(57,59);(1,4)(2,9)(3,15)(5,24)(6,29)(7,20)(8,31)(10,38)(11,35)(12,43)(13,37)(14,44)(16,49)(17,47)(18,52)(19,28)(21,25)(22,30)(23,48)(26,56)(27,36)(32,61)(33,60)(34,42)(39,63)(40,46)(41,64)(45,62)(50,54)(51,53)(55,57)(58,59);(2,10)(4,20)(5,25)(8,14)(9,35)(11,13)(12,39)(15,47)(16,23)(19,52)(22,29)(26,59)(27,41)(31,46)(32,45)(33,40)(34,62)(36,63)(37,38)(42,51)(43,64)(44,60)(50,57)(53,61)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312282, #72312329, #72312379, … |
| maximal normal subgroup of | #72312282, #72312329, #72312379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407081);