Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24);(5,6)(7,8)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23);(10,15)(14,21)(16,19)(22,23);(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24);(1,3,4,2)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24);(9,12)(10,16)(11,17)(13,20)(14,22)(15,19)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=i,b^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
| maximal subgroup of | #72343397, #72343789, C4×#62842 |
| maximal normal subgroup of | #72343397, #72343789, C4×#62842 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532936);