Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,11)(7,12,8,10)(13,14)(15,23)(16,27)(17,31)(18,24)(19,20)(21,28)(22,34)(25,38)(26,36)(29,44)(30,35)(32,43)(33,40)(37,41)(39,42);(2,3)(5,6)(7,10)(8,12)(9,11)(13,15,18,19)(14,20,24,23)(16,28,30,25)(17,26,34,33)(21,27,38,35)(22,36,31,40)(29,42,32,41)(37,43,39,44);(1,2,4,3)(13,16,15,25,18,30,19,28)(14,21,20,35,24,38,23,27)(17,32,26,42,34,29,33,41)(22,39,36,43,31,37,40,44);(5,7,9,8)(6,10,11,12)(13,17)(14,22)(15,26)(16,29)(18,34)(19,33)(20,36)(21,37)(23,40)(24,31)(25,41)(27,43)(28,42)(30,32)(35,44)(38,39);(1,3,4,2)(5,8,9,7)(6,10,11,12)(13,15,18,19)(14,23,24,20)(16,25,30,28)(17,33,34,26)(21,27,38,35)(22,36,31,40)(29,42,32,41)(37,44,39,43)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724236);