Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
3575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,7)(8,10,12,11)(13,14)(15,16)(17,19)(18,20)(22,23)(24,28);(1,2,3,4)(6,7)(10,11)(13,15)(14,18)(16,19)(17,20)(21,22,25,23)(24,26,28,27);(1,2)(3,4)(8,12)(10,11)(14,19)(16,18)(21,22,25,23)(24,27,28,26);(5,7,9,6)(8,10,12,11)(13,15)(14,16)(17,20)(18,19)(21,23,25,22)(24,26,28,27);(5,8,9,12)(6,10,7,11)(13,16)(14,20)(15,19)(17,18)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,c^2=i,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72326953, #72326969, #72327019, … |
| maximal normal subgroup of | #72326953, #72326969, #72327019, … |
| maximal quotient of | #72330336 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3471570);