Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 |
| prime radical | C2×#3212 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2009 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)(7,11)(8,10)(9,12)(13,14,18,24)(15,25,29,20)(16,31,33,22)(17,23,19,35)(21,34,40,32)(26,41,27,37)(28,42,30,38)(36,43,39,44);(2,3)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,27)(17,28)(18,29)(19,30)(21,39)(22,37)(23,38)(24,25)(26,33)(31,41)(32,44)(34,43)(35,42)(36,40);(1,2,4,3)(7,12)(9,11)(13,16)(14,21)(15,26)(17,32)(18,33)(19,34)(20,36)(22,23)(24,40)(25,39)(27,29)(28,43)(30,44)(31,35)(37,42)(38,41);(6,10)(9,11)(13,16)(14,22)(15,27)(17,34)(18,33)(19,32)(20,37)(21,23)(24,31)(25,41)(26,29)(28,43)(30,44)(35,40)(36,42)(38,39);(1,3,4,2)(13,17,18,19)(14,23,24,35)(15,28,29,30)(16,32,33,34)(20,38,25,42)(21,22,40,31)(26,43,27,44)(36,41,39,37)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717748);