Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| 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 | C2×#3006 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3040 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)(7,11)(8,10)(9,12)(15,23)(16,17)(19,26)(21,22)(24,25)(27,28);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,14,18,20)(15,19,24,27)(16,21,17,22)(23,26,25,28);(7,12)(9,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,27)(21,26)(22,28);(6,10)(9,11)(14,20)(19,27)(21,22)(26,28);(13,16,18,17)(14,21,20,22)(15,23,24,25)(19,26,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239649, #72282304, #72282339, … |
| maximal normal subgroup of | #72239649, #72282304, #72282339, … |
| maximal quotient of | #72239718, #72313722 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282713);