Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | |
1662 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)(17,18,23,29,24,30,22,28)(19,21,27,39,33,40,32,25)(20,34,37,26,38,43,36,44)(31,45,42,48,47,41,46,35);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,31)(21,28)(22,27)(23,32)(24,33)(26,41)(29,40)(30,39)(34,35)(36,42)(37,46)(38,47)(43,48)(44,45);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,22,36,24,38,23,37)(18,26,29,43,30,44,28,34)(19,31,27,42,33,47,32,46)(21,35,25,41,40,48,39,45);(3,9)(6,13)(10,15)(14,16)(17,21,22,25,24,40,23,39)(18,27,28,19,30,32,29,33)(20,35,36,41,38,48,37,45)(26,42,34,31,44,46,43,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,24,23)(18,28,30,29)(19,27,33,32)(20,36,38,37)(21,39,40,25)(26,34,44,43)(31,42,47,46)(35,45,48,41)Polycyclic presentation:
2^9|a^2=fh,c^2=ghi,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104027);