Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,17)(18,20)(22,24)(23,27);(6,9)(7,12)(13,14,16,17)(15,19,20,18)(22,24)(26,28);(6,9)(10,11)(14,17)(18,19)(21,22,25,24)(23,28,27,26);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,23,25,27)(22,26,24,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72327945, #72328008, #72328049 |
| maximal normal subgroup of | #72327945, #72328008, #72328049 |
| maximal quotient of | #72330092 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5120713);