Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14,18,24,19,25,17,23)(15,21,27,35,28,37,32,20)(16,22,30,38,31,39,33,40)(26,36,29,43,42,44,41,34);(6,9)(7,11)(13,15)(14,20)(16,29)(17,27)(18,32)(19,28)(21,23)(22,36)(24,37)(25,35)(26,30)(31,41)(33,42)(34,38)(39,44)(40,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,26)(17,32)(18,27)(19,28)(20,23)(22,36)(24,35)(25,37)(29,30)(31,42)(33,41)(34,40)(38,43)(39,44);(6,9)(10,12)(13,16,18,30,19,31,17,33)(14,22,24,38,25,39,23,40)(15,26,27,29,28,42,32,41)(20,34,21,36,35,43,37,44);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,27,28,32)(16,30,31,33)(20,21,35,37)(22,38,39,40)(26,41,42,29)(34,44,43,36)Polycyclic presentation:
2^9|a^2=f,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053292);