Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2227 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)(13,14,20,26)(15,27,31,42)(16,22,17,23)(18,25,19,24)(21,35,38,30)(28,37,32,44)(29,36,43,39)(33,41,34,40);(6,9)(7,11)(13,15,18,35,20,31,19,30)(14,21,24,42,26,38,25,27)(16,32,34,29,17,28,33,43)(22,39,41,37,23,36,40,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,28)(17,20)(18,34)(19,33)(21,36)(23,26)(24,41)(25,40)(27,37)(29,30)(31,32)(35,43)(38,39)(42,44);(6,9)(10,12)(13,17,20,16)(14,23,26,22)(15,29,31,43)(18,34,19,33)(21,37,38,44)(24,41,25,40)(27,36,42,39)(28,35,32,30);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,26,25)(15,30,31,35)(16,33,17,34)(21,27,38,42)(22,40,23,41)(28,43,32,29)(36,44,39,37)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050957);