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×#3497 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
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)(15,22,29,35)(16,30,33,38)(17,23,18,24)(20,27,36,28)(21,32,39,31)(26,40,43,34)(37,41,44,42);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,26)(17,27)(18,28)(19,29)(21,34)(22,23)(24,35)(25,36)(30,44)(31,41)(32,42)(33,43)(37,38)(39,40);(7,11)(9,12)(13,16,19,33)(14,21,25,39)(15,26,29,43)(17,31,18,32)(20,34,36,40)(22,37,35,44)(23,38,24,30)(27,41,28,42);(7,11)(9,12)(13,15)(14,22)(16,26)(17,28)(18,27)(19,29)(20,23)(21,37)(24,36)(25,35)(30,40)(31,42)(32,41)(33,43)(34,38)(39,44);(6,10)(9,12)(13,17,19,18)(14,23,25,24)(15,27,29,28)(16,31,33,32)(20,22,36,35)(21,38,39,30)(26,41,43,42)(34,37,40,44)Polycyclic presentation:
2^9|a^2=i,c^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,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, 6377396);