Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,8,11,10)(13,14,19,33,18,24,17,23)(15,20,31,21,26,35,27,36)(16,28,32,39,30,38,29,22)(25,40,43,44,41,37,42,34);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15)(14,20)(16,25)(17,27)(18,26)(19,31)(21,33)(22,34)(23,36)(24,35)(28,40)(29,42)(30,41)(32,43)(37,38)(39,44);(2,4)(5,8,9,10)(6,11,12,7)(13,15)(14,21)(16,25)(17,31)(18,26)(19,27)(20,33)(22,37)(23,35)(24,36)(28,44)(29,43)(30,41)(32,42)(34,38)(39,40);(6,12)(7,11)(13,16,18,30)(14,22,24,39)(15,25,26,41)(17,32,19,29)(20,34,35,44)(21,37,36,40)(23,28,33,38)(27,43,31,42);(15,26)(20,35)(21,36)(25,41)(27,31)(34,44)(37,40)(42,43)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411017);