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 | |
1814 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)(13,14,20,34,19,24,18,23)(15,25,17,21,28,36,27,37)(16,29,31,39,33,22,32,38)(26,41,30,44,43,40,42,35);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,24,37)(16,26,33,43)(17,20,27,18)(22,35,29,44)(23,36,34,25)(30,31,42,32)(38,40,39,41);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,26)(17,30)(18,32)(19,33)(20,31)(21,35)(23,39)(24,29)(25,40)(27,42)(28,43)(34,38)(36,41)(37,44);(7,12)(9,11)(13,17,18,15,19,27,20,28)(14,21,23,25,24,37,34,36)(16,30,32,26,33,42,31,43)(22,35,39,40,29,44,38,41);(6,10)(9,11)(13,18,19,20)(14,23,24,34)(15,17,28,27)(16,31,33,32)(21,36,37,25)(22,38,29,39)(26,42,43,30)(35,40,44,41)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=i,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, 6971988);