Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021351 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,22,27,23,28,39,26)(19,29,31,43,33,42,32,24)(20,34,21,25,36,45,38,46)(30,44,37,48,35,40,47,41);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,24)(20,35)(21,37)(22,32)(23,33)(25,44)(26,29)(27,42)(28,43)(30,36)(31,39)(34,48)(38,47)(40,46)(41,45);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,22,21,23,36,39,38)(18,25,26,34,28,46,27,45)(19,30,32,47,33,35,31,37)(24,40,29,41,43,44,42,48);(4,12)(7,16)(9,11)(13,15)(17,21,22,36,23,38,39,20)(18,25,27,45,28,46,26,34)(19,30,31,37,33,35,32,47)(24,41,29,44,43,48,42,40);(2,8)(3,10)(7,16)(9,11)(17,22,23,39)(18,26,28,27)(19,31,33,32)(20,21,36,38)(24,42,43,29)(25,34,46,45)(30,37,35,47)(40,48,44,41)Polycyclic presentation:
2^9|a^2=fhi,c^2=g,d^2=f,e^2=i,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=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949270);