Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1910 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)(7,11)(10,12)(13,14,20,26,19,25,18,24)(15,27,31,38,30,21,29,39)(16,23,32,40,17,33,35,22)(28,37,43,36,34,41,44,42);(1,2,3,4)(6,9)(7,11)(13,15)(14,21)(16,28)(17,34)(18,29)(19,30)(20,31)(22,36)(23,41)(24,38)(25,27)(26,39)(32,43)(33,37)(35,44)(40,42);(5,6)(7,10)(8,9)(11,12)(13,16,19,17)(14,22,25,40)(15,28,30,34)(18,32,20,35)(21,36,27,42)(23,26,33,24)(29,43,31,44)(37,38,41,39);(2,4)(6,9)(10,12)(13,17,19,16)(14,23,25,33)(15,28,30,34)(18,32,20,35)(21,37,27,41)(22,26,40,24)(29,44,31,43)(36,38,42,39);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,24,25,26)(15,29,30,31)(16,32,17,35)(21,38,27,39)(22,23,40,33)(28,43,34,44)(36,41,42,37)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7056330);