Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1844 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)(13,14,20,17,19,24,18,23)(15,25,27,21,29,35,28,36)(16,22,31,30,33,38,32,37)(26,39,42,34,41,44,40,43);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,21,24,36)(16,26,33,41)(17,25,23,35)(18,27,20,28)(22,34,38,43)(30,39,37,44)(31,40,32,42);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,26)(17,30)(18,32)(19,33)(20,31)(21,34)(23,37)(24,38)(25,39)(27,42)(28,40)(29,41)(35,44)(36,43);(7,12)(9,11)(13,17,20,24,19,23,18,14)(15,21,27,35,29,36,28,25)(16,30,31,38,33,37,32,22)(26,34,42,44,41,43,40,39);(6,10)(9,11)(13,18,19,20)(14,23,24,17)(15,27,29,28)(16,31,33,32)(21,35,36,25)(22,30,38,37)(26,40,41,42)(34,39,43,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=h,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972230);