Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18)(19,29)(20,33)(21,32)(22,41)(23,27)(24,26)(25,37)(28,42)(30,39)(31,40)(34,35)(36,44)(38,45)(43,48)(46,47);(3,9)(4,12)(6,13)(7,11)(17,19)(18,24)(20,34)(21,27)(22,32)(23,31)(25,39)(26,42)(28,40)(29,41)(30,38)(33,48)(35,44)(36,46)(37,43)(45,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,25)(19,30)(21,39)(22,36)(23,38)(24,35)(26,46)(27,44)(28,45)(29,47)(31,34)(32,43)(33,41)(37,42)(40,48);(3,9)(6,13)(10,15)(14,16)(17,21)(18,26)(19,28)(20,35)(22,29)(23,24)(25,43)(27,32)(30,33)(31,41)(34,45)(36,48)(37,47)(38,39)(40,42)(44,46);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(19,31)(25,44)(26,32)(28,41)(30,34)(33,45)(43,46)Polycyclic presentation:
2^9|a^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7073048);