Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1840 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)(14,20)(15,24)(17,18)(21,35)(22,28)(23,32)(25,34)(26,39)(29,33)(36,41)(37,38)(40,44);(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,25,24,21)(16,28,30,37)(17,20,18,32)(22,29,38,33)(26,41,39,44)(27,34,31,35)(36,42,40,43);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,24)(14,21,23,25)(16,26,30,39)(17,31,18,27)(20,34,32,35)(22,36,38,40)(28,44,37,41)(29,43,33,42);(7,12)(9,11)(13,16)(14,22)(15,26)(17,29)(18,33)(19,30)(20,28)(21,36)(23,38)(24,39)(25,40)(27,42)(31,43)(32,37)(34,44)(35,41);(6,10)(9,11)(13,17,19,18)(14,20,23,32)(15,27,24,31)(16,29,30,33)(21,35,25,34)(22,28,38,37)(26,42,39,43)(36,41,40,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970276);