Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10)(11,18)(12,21)(13,15)(14,24)(16,20)(17,23)(19,22);(1,2)(3,4)(5,6)(7,8)(10,15)(12,16)(17,24)(18,22);(6,8)(9,11)(10,16)(12,15)(13,19)(14,21)(17,22)(18,24)(20,23);(2,4)(9,12,13,16)(10,11,15,19)(14,18,23,22)(17,21,24,20);(10,15)(11,19)(17,24)(20,21)Polycyclic presentation:
2^9|a^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307418, #72307429, #72307451, … |
| maximal normal subgroup of | #72307418, #72307429, #72307451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408168);