Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,25)(15,26,30,36)(16,22,33,38)(17,23,18,24)(20,29,37,28)(21,32,39,31)(27,41,44,34)(35,42,40,43);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,30)(14,20,25,37)(16,27,33,44)(17,29,18,28)(21,34,39,41)(22,35,38,40)(23,26,24,36)(31,43,32,42);(7,11)(9,12)(13,16)(14,21)(15,27)(17,31)(18,32)(19,33)(20,34)(22,24)(23,38)(25,39)(26,40)(28,42)(29,43)(30,44)(35,36)(37,41);(7,11)(9,12)(13,16)(14,22)(15,27)(17,32)(18,31)(19,33)(20,35)(21,24)(23,39)(25,38)(26,41)(28,43)(29,42)(30,44)(34,36)(37,40);(6,10)(9,12)(13,17,19,18)(14,23,25,24)(15,28,30,29)(16,31,33,32)(20,36,37,26)(21,38,39,22)(27,42,44,43)(34,35,41,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377412);