Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1812 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)(13,14,18,23,19,22,33,17)(15,24,27,36,26,39,34,20)(16,21,31,29,30,38,44,32)(25,40,41,37,28,42,43,35);(5,6)(7,11)(8,10)(9,12)(13,15,19,26)(14,20,22,36)(16,28,30,25)(17,24,23,39)(18,34,33,27)(21,37,38,35)(29,40,32,42)(31,41,44,43);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,30)(14,21,22,38)(15,25,26,28)(17,32,23,29)(18,31,33,44)(20,35,36,37)(24,40,39,42)(27,41,34,43);(7,12)(9,11)(13,17,33,22,19,23,18,14)(15,20,34,39,26,36,27,24)(16,29,44,21,30,32,31,38)(25,37,43,40,28,35,41,42);(6,10)(9,11)(14,22)(15,26)(16,30)(17,23)(27,34)(31,44)(35,37)(40,42)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6971989);