Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×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×#3151 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2802 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)(13,14)(15,17)(16,23)(18,22)(19,26)(20,24)(21,27)(25,28);(1,3,4,2)(5,7,10,6)(8,11,9,12)(13,15,17,14)(16,19,25,21)(18,22,24,20)(23,27,28,26);(5,8,10,9)(6,11,7,12)(13,16,24,27,17,25,18,26)(14,19,22,28,15,21,20,23);(5,9,10,8)(6,11,7,12)(14,15)(19,21)(20,22)(23,28);(16,25)(19,21)(23,28)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240343, #72240346, #72240751, … |
| maximal normal subgroup of | #72240343, #72240346, #72240751, … |
| maximal quotient of | #72240759, #72240760, #72240767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283638);