Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432953 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3021 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)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,18,22,26)(19,23,29,37)(20,24,32,40)(21,33,34,25)(27,38,30,35)(28,42,43,36)(31,45,46,39)(41,47,44,48);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,38)(25,36)(26,37)(27,32)(31,44)(33,42)(34,43)(35,40)(39,48)(41,46)(45,47);(3,9)(6,12)(10,15)(13,16)(17,20)(18,24)(19,27)(21,31)(22,32)(23,35)(25,39)(26,40)(28,41)(29,30)(33,45)(34,46)(36,47)(37,38)(42,48)(43,44);(2,7)(6,12)(8,14)(13,16)(17,21,22,34)(18,25,26,33)(19,28,29,43)(20,31,32,46)(23,36,37,42)(24,39,40,45)(27,41,30,44)(35,47,38,48);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)(21,34)(25,33)(28,43)(31,46)(36,42)(39,45)(41,44)(47,48)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=hi,b^a=bf,c^a=cg,d^a=dh,c^b=ch,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343843);