Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,20)(15,24,25,39)(16,29,31,22)(17,23,19,33)(21,27,34,28)(26,41,40,38)(30,37,32,36)(35,42,44,43);(2,3)(5,7,8,12)(6,9,10,11)(14,20)(15,25)(22,29)(23,33)(26,40)(27,28)(36,37)(42,43);(1,2,4,3)(6,10)(7,12)(13,15,17,27,18,25,19,28)(14,21,33,39,20,34,23,24)(16,26,32,43,31,40,30,42)(22,35,36,41,29,44,37,38);(6,10)(9,11)(13,16,18,31)(14,22,20,29)(15,26,25,40)(17,32,19,30)(21,35,34,44)(23,37,33,36)(24,38,39,41)(27,43,28,42);(1,3,4,2)(13,17,18,19)(14,23,20,33)(15,27,25,28)(16,30,31,32)(21,24,34,39)(22,36,29,37)(26,42,40,43)(35,41,44,38)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723570);