Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4219 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)(7,10,11,12)(13,14,18,20)(15,16,17,19)(22,26)(23,28);(6,9)(10,12)(13,15,18,17)(14,19,20,16)(21,22)(23,24)(25,26)(27,28);(7,11)(10,12)(14,20)(15,17)(22,26)(24,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,27)(24,26)(25,28);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,16,20,19)(21,24,25,27)(22,23,26,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72234000, #72234023, #72234030, … |
| maximal normal subgroup of | #72234000, #72234023, #72234030, … |
| maximal quotient of | #72329914 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1984826);