Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(14,18)(16,19)(20,24)(21,25)(23,26)(27,28);(1,3,4,2)(5,7,10,6)(8,11,9,12);(13,14,19,25,17,21,16,18)(15,20,26,28,22,27,23,24);(5,8,10,9)(6,11,7,12)(13,15)(14,20)(16,23)(17,22)(18,24)(19,26)(21,27)(25,28);(5,9,10,8)(6,11,7,12)(14,21)(15,22)(18,25)(23,26)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=gi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^c=ei,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72240753, #72257158, #72257196, … |
| maximal normal subgroup of | #72240753, #72257158, #72257196, … |
| maximal quotient of | #72240784 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6297092);