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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1808 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)(13,14,18,23,19,22,34,17)(15,24,27,37,26,40,35,20)(16,28,32,33,31,21,38,30)(25,42,43,36,29,41,44,39);(5,6)(7,11)(8,10)(9,12)(13,15,19,26)(14,20,22,37)(16,29,31,25)(17,24,23,40)(18,35,34,27)(21,39,28,36)(30,41,33,42)(32,43,38,44);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,31)(14,21,22,28)(15,25,26,29)(17,33,23,30)(18,32,34,38)(20,36,37,39)(24,41,40,42)(27,43,35,44);(7,12)(9,11)(13,17,34,22,19,23,18,14)(15,20,35,40,26,37,27,24)(16,30,38,21,31,33,32,28)(25,39,44,41,29,36,43,42);(6,10)(9,11)(14,22)(15,26)(16,31)(17,23)(27,35)(32,38)(36,39)(41,42)Polycyclic presentation:
2^9|a^2=fgi,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, 6972005);