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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2430 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)(13,14,19,24)(15,20,27,36)(16,28,30,37)(17,23,18,21)(22,32,38,29)(25,39,42,44)(26,33,31,35)(34,40,43,41);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,19,27)(14,20,24,36)(16,25,30,42)(17,31,18,26)(21,33,23,35)(22,34,38,43)(28,39,37,44)(29,41,32,40);(2,4)(5,8,9,12)(6,10,7,11)(14,21)(17,18)(20,33)(22,28)(23,24)(26,31)(29,32)(34,39)(35,36)(37,38)(40,41)(43,44);(6,7)(10,11)(13,16)(14,22)(15,25)(17,29)(18,32)(19,30)(20,34)(21,28)(23,37)(24,38)(26,40)(27,42)(31,41)(33,39)(35,44)(36,43);(13,17,19,18)(14,23,24,21)(15,26,27,31)(16,29,30,32)(20,33,36,35)(22,37,38,28)(25,40,42,41)(34,39,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^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,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, 7408990);