Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3552 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14,18,22,19,24,17,23)(15,20,26,36,27,35,28,34)(16,29,30,21,32,37,31,38)(25,39,42,33,41,43,40,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,28)(18,26)(19,27)(21,33)(22,36)(23,34)(24,35)(29,39)(30,42)(31,40)(32,41)(37,43)(38,44);(1,2,3,4)(13,16)(14,21)(15,25)(17,30)(18,31)(19,32)(20,33)(22,29)(23,37)(24,38)(26,40)(27,41)(28,42)(34,43)(35,44)(36,39);(6,9)(10,12)(13,17,19,18)(14,22,24,23)(15,26,27,28)(16,30,32,31)(20,34,35,36)(21,29,38,37)(25,40,41,42)(33,43,44,39);(5,7,8,11)(6,10,9,12)(15,27)(20,35)(25,41)(26,28)(33,44)(34,36)(39,43)(40,42)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=dfi,e^b=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300972);