Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623308 |
| 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 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1301 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)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,23,28,36)(20,24,33,39)(21,34,32,27)(26,41,40,31)(29,35,42,46)(30,45,43,38)(37,48,47,44);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,35)(25,36)(26,37)(27,38)(31,44)(32,43)(33,42)(34,45)(39,46)(40,47)(41,48);(3,10)(6,9)(11,16)(13,15)(18,24)(20,31)(21,32)(23,35)(25,39)(26,27)(29,44)(30,43)(33,41)(34,40)(36,46)(37,38)(42,48)(45,47);(2,7)(6,9)(8,14)(13,15)(19,28)(23,36)(29,42)(30,43)(35,46)(37,47)(38,45)(44,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(18,25)(23,36)(24,39)(26,40)(27,34)(35,46)(37,47)(38,45)Polycyclic presentation:
2^9|a^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,d^b=di,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286755, #72286841, #72286904, … |
| maximal normal subgroup of | #72286755, #72286841, #72286904, … |
| maximal quotient of | #72343325, #72343370, #72343459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530725);