Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #62325 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,20)(11,16,22,35)(12,25,27,52)(13,30,32,58)(14,33,31,19)(17,38,39,28)(18,41,42,26)(21,43,47,67)(23,50,48,71)(24,51,49,37)(29,57,56,54)(34,59,61,45)(36,63,62,44)(40,55,53,64)(46,70,69,66)(60,68,65,72);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,34)(18,36)(19,37)(20,35)(25,43)(26,44)(27,47)(28,45)(29,46)(30,50)(31,49)(32,48)(33,51)(38,59)(39,61)(40,60)(41,63)(42,62)(52,67)(53,65)(54,66)(55,68)(56,69)(57,70)(58,71)(64,72);(9,12,26,55,14,29,32,38,15,27,41,64,31,56,13,28)(10,17,30,54,19,40,42,52,20,39,58,57,33,53,18,25)(11,21,44,68,24,46,48,59,22,47,63,72,49,69,23,45)(16,34,50,66,37,60,62,67,35,61,71,70,51,65,36,43);(3,8)(5,7)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,34)(18,36)(19,37)(20,35)(25,43)(26,44)(27,47)(28,45)(29,46)(30,50)(31,49)(32,48)(33,51)(38,59)(39,61)(40,60)(41,63)(42,62)(52,67)(53,65)(54,66)(55,68)(56,69)(57,70)(58,71)(64,72);(2,6)(5,7)(11,22)(12,27)(16,35)(17,39)(23,48)(24,49)(25,52)(28,38)(29,56)(36,62)(37,51)(40,53)(44,63)(50,71)(54,57)(55,64)Polycyclic presentation:
2^9|a^2=i,c^2=gh,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532085);