Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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×#3153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,27)(17,19)(18,31)(20,26)(21,33)(22,30)(24,40)(25,38)(28,37)(29,35)(32,34)(36,41)(39,43)(42,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(14,19)(16,28)(20,34)(22,31)(23,38)(24,41)(29,33)(42,43);(2,4)(5,8,9,12)(6,10,7,11)(13,15,17,25)(14,20,19,34)(16,24,28,41)(18,32,30,26)(21,36,35,40)(22,38,31,23)(27,44,37,39)(29,43,33,42);(6,7)(10,11)(13,16,17,28)(14,21,19,35)(15,24,25,41)(18,33,30,29)(20,36,34,40)(22,27,31,37)(23,39,38,44)(26,43,32,42);(14,19)(15,25)(21,35)(22,31)(24,41)(26,32)(27,37)(42,43)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=i,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^c=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, 7408977);