Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,30)(21,28)(22,37)(23,27)(24,32)(25,33)(29,39)(31,41)(34,44)(35,46)(36,40)(38,45)(42,48)(43,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,34)(21,32)(22,33)(23,31)(25,40)(26,28)(27,44)(29,42)(30,41)(35,48)(36,47)(37,43)(38,39)(45,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,25)(20,31)(21,32)(22,38)(23,34)(24,40)(26,37)(27,42)(28,43)(29,44)(30,45)(33,39)(35,47)(36,48)(41,46);(3,11)(6,9)(10,16)(13,15)(17,19)(18,26)(20,34)(21,32)(22,33)(23,31)(24,28)(25,37)(27,41)(29,45)(30,44)(35,48)(36,47)(38,39)(40,43)(42,46);(2,8)(6,9)(7,14)(13,15)(17,20,21,23)(18,27,28,30)(19,31,32,34)(22,35,36,39)(24,41,26,44)(25,42,43,45)(29,40,46,37)(33,38,47,48)Polycyclic presentation:
2^9|c^2=fi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72318797, #72320335, #72320368, … |
| maximal normal subgroup of | #72318797, #72320335, #72320368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531552);