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×#3154 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2432 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)(15,19)(16,27)(17,20)(18,23)(21,28)(22,30)(24,38)(25,34)(26,32)(29,37)(31,36)(33,40)(35,41)(39,44)(42,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,24)(17,26)(18,25)(20,32)(21,33)(22,35)(23,34)(27,38)(28,40)(29,39)(30,41)(31,42)(36,43)(37,44);(2,4)(5,8,9,12)(6,10,7,11)(14,20)(17,30)(19,32)(21,27)(22,23)(26,41)(28,31)(33,38)(34,35)(36,37)(40,42)(43,44);(6,7)(10,11)(13,16,18,29)(14,21,23,36)(15,24,25,39)(17,31,30,28)(19,33,34,43)(20,27,22,37)(26,42,41,40)(32,38,35,44);(15,25)(19,34)(24,39)(26,41)(32,35)(33,43)(38,44)(40,42)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7408931);