Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,20,24)(15,27,31,39)(16,32,34,41)(17,23,33,40)(18,26,19,25)(21,30,38,29)(22,35,42,28)(36,44,37,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,29,20,31,19,30)(14,21,25,27,24,38,26,39)(16,33,35,37,34,17,28,36)(22,40,41,44,42,23,32,43);(6,9)(10,12)(13,16,18,35,20,34,19,28)(14,22,25,41,24,42,26,32)(15,17,29,36,31,33,30,37)(21,23,27,43,38,40,39,44);(6,9)(10,12)(13,17)(14,23)(15,28)(16,29)(18,36)(19,37)(20,33)(21,32)(22,27)(24,40)(25,43)(26,44)(30,34)(31,35)(38,41)(39,42);(5,7,8,11)(6,10,9,12)(14,24)(16,34)(17,33)(21,38)(25,26)(27,39)(28,35)(36,37)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,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, 6401620);