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×#3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4210 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)(13,14,19,24)(15,25,29,33)(16,22,32,38)(17,20,18,23)(21,28,36,27)(26,39,42,43)(30,37,31,34)(35,40,44,41);(2,4)(5,7)(6,9)(8,11)(10,12)(14,20)(17,18)(21,33)(22,34)(23,24)(25,36)(27,28)(30,31)(35,43)(37,38)(39,44)(40,41);(7,11)(9,12)(13,15,19,29)(14,21,24,36)(16,26,32,42)(17,27,18,28)(20,33,23,25)(22,35,38,44)(30,40,31,41)(34,43,37,39);(7,11)(9,12)(13,16)(14,22)(15,26)(17,31)(18,30)(19,32)(20,34)(21,35)(23,37)(24,38)(25,39)(27,41)(28,40)(29,42)(33,43)(36,44);(6,10)(9,12)(13,17,19,18)(14,20,24,23)(15,27,29,28)(16,30,32,31)(21,33,36,25)(22,37,38,34)(26,40,42,41)(35,39,44,43)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,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, 6377414);