Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,25,20,26,18,24)(15,21,30,38,31,39,29,37)(16,32,33,40,17,23,34,22)(27,41,42,44,28,36,43,35);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,20,31)(14,21,26,39)(16,27,17,28)(18,29,19,30)(22,35,40,44)(23,36,32,41)(24,37,25,38)(33,42,34,43);(5,7)(6,9)(8,12)(10,11)(13,16,20,17)(14,22,26,40)(15,27,31,28)(18,33,19,34)(21,35,39,44)(23,24,32,25)(29,42,30,43)(36,37,41,38);(1,3,4,2)(7,12)(9,11)(13,17,20,16)(14,23,26,32)(15,28,31,27)(18,33,19,34)(21,36,39,41)(22,24,40,25)(29,42,30,43)(35,37,44,38);(6,10)(9,11)(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,33,17,34)(21,37,39,38)(22,32,40,23)(27,42,28,43)(35,41,44,36)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913410);