Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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×#3264 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2532 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)(8,9)(11,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,26)(24,28)(25,27);(6,9)(7,11);(14,19)(16,17)(20,26)(21,22)(23,25)(27,28);(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,27)(22,28);(1,3,4,2)(6,9)(10,12)(13,16,18,17)(14,21,22,19)(15,23,24,25)(20,26,28,27)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^c=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72283581, #72283762, #72283850, … |
| maximal normal subgroup of | #72283581, #72283762, #72283850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306763);