Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C4 | C24×C8 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#576 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10,13,15)(11,16,18,22)(12,14,20,17)(19,21,24,23);(10,14)(12,20)(15,17)(16,21)(19,24)(22,23);(1,3,4,8,5,2,6,7)(10,15)(14,17)(16,22)(21,23);(9,11)(10,16)(12,19)(13,18)(14,21)(15,22)(17,23)(20,24);(11,18)(16,22)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=g,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fh,e^d=eh164 conjugacy classes — view the full character table.
| maximal subgroup of | #72216435, #72216441, #72216443, … |
| maximal normal subgroup of | #72216435, #72216441, #72216443, … |
| maximal quotient of | D4×#1580, C2×#6344533, #72287023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251729);