Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,31,42)(16,22,17,23)(18,25,19,24)(21,35,38,30)(28,43,32,37)(29,39,44,36)(33,41,34,40);(5,6)(7,11)(8,10)(9,12)(13,15,20,31)(14,21,26,38)(16,32,17,28)(18,30,19,35)(22,39,23,36)(24,27,25,42)(29,33,44,34)(37,40,43,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,17)(14,22,26,23)(15,28,31,32)(18,34,19,33)(21,36,38,39)(24,41,25,40)(27,43,42,37)(29,30,44,35);(7,12)(9,11)(13,17,20,16)(14,23,26,22)(15,29,31,44)(18,34,19,33)(21,37,38,43)(24,41,25,40)(27,39,42,36)(28,30,32,35);(6,10)(9,11)(13,18,20,19)(14,24,26,25)(15,30,31,35)(16,33,17,34)(21,27,38,42)(22,40,23,41)(28,29,32,44)(36,37,39,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dhi,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, 6901366);