Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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×#3191 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,24,19,20)(16,27,30,35)(17,31,33,22)(21,25,38,28)(26,36,34,40)(29,44,32,43)(37,42,39,41);(1,2)(3,4)(6,9)(7,11)(13,15,18,19)(14,20,23,24)(16,28,30,25)(17,26,33,34)(21,27,38,35)(22,36,31,40)(29,42,32,41)(37,43,39,44);(2,4)(13,16,15,25,18,30,19,28)(14,21,20,35,23,38,24,27)(17,32,26,42,33,29,34,41)(22,39,36,43,31,37,40,44);(6,9)(10,12)(13,17)(14,22)(15,26)(16,29)(18,33)(19,34)(20,36)(21,37)(23,31)(24,40)(25,41)(27,43)(28,42)(30,32)(35,44)(38,39);(2,4)(5,7,8,11)(6,10,9,12)(16,30)(17,33)(21,38)(22,31)(25,28)(26,34)(27,35)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6592663);