Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,18)(11,15,12,16)(13,17,21,24)(19,22,20,23);(1,2,3,4)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23);(6,8)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24);(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22);(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72308822, #72308978, #72309052 |
| maximal normal subgroup of | #72308822, #72308978, #72309052 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481043);