Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14,18,22)(15,19,25,35)(16,27,29,37)(17,21,31,23)(20,30,36,28)(24,38,39,44)(26,33,41,34)(32,42,43,40);(1,2)(3,4)(6,9)(7,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,32)(21,33)(22,35)(23,34)(27,38)(28,40)(29,39)(30,42)(31,41)(36,43)(37,44);(2,4)(13,16,18,29)(14,20,22,36)(15,24,25,39)(17,30,31,28)(19,32,35,43)(21,27,23,37)(26,42,41,40)(33,38,34,44);(5,7,8,12)(6,10,9,11)(14,21)(17,31)(19,33)(20,27)(22,23)(26,41)(28,30)(32,38)(34,35)(36,37)(40,42)(43,44);(14,22)(15,25)(20,36)(21,23)(24,39)(26,41)(27,37)(40,42)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409510);