Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2058 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)(13,14,20,26)(15,27,29,39)(16,32,34,40)(17,23,35,42)(18,25,19,24)(21,31,38,30)(22,33,41,28)(36,44,37,43);(5,7)(6,9)(8,11)(10,12)(13,15,19,31,20,29,18,30)(14,21,25,39,26,38,24,27)(16,17,28,37,34,35,33,36)(22,23,32,44,41,42,40,43);(2,4)(7,11)(9,12)(13,16,18,33,20,34,19,28)(14,22,24,40,26,41,25,32)(15,17,30,36,29,35,31,37)(21,23,27,43,38,42,39,44);(7,11)(9,12)(13,17)(14,23)(15,28)(16,30)(18,36)(19,37)(20,35)(21,32)(22,27)(24,43)(25,44)(26,42)(29,33)(31,34)(38,40)(39,41);(6,10)(9,12)(15,29)(17,35)(21,38)(23,42)(27,39)(30,31)(36,37)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,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,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378082);