Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(15,22)(16,25)(19,27)(20,21)(23,26);(13,14)(15,23)(16,20)(17,21)(18,25)(19,22)(24,26)(27,28);(6,9)(10,12)(13,15)(14,19)(16,24)(17,22)(18,26)(20,27)(21,23)(25,28);(14,20)(16,25)(18,21)(19,27)(23,26)(24,28);(5,7,8,11)(6,10,9,12)(14,21)(15,22)(18,20)(24,28)Polycyclic presentation:
2^9|a^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72240755, #72256930, #72256969 |
| maximal normal subgroup of | #72240755, #72256930, #72256969 |
| maximal quotient of | #72240762, #72240778, #72264214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288173);