Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(14,15)(18,19)(21,22,25,27)(23,24,28,26);(2,4)(6,9)(10,12)(13,14,17,15)(16,19,20,18)(21,23,25,28)(22,26,27,24);(1,2)(3,4)(7,11)(10,12)(14,15)(18,19)(22,27)(24,26);(5,7,8,11)(6,10,9,12)(13,15,17,14)(16,19,20,18)(21,24)(22,23)(25,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72300475, #72300478, #72300573, … |
| maximal normal subgroup of | #72300475, #72300478, #72300573, … |
| maximal quotient of | #72331324, #72333133, #72333194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 602738);