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 | |
4439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,19)(18,20)(21,23,26,25)(22,27,24,28);(1,2)(3,4)(7,11)(10,12)(14,17)(15,20)(21,22,26,24)(23,28,25,27);(5,6)(7,12)(8,9)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24,26,22)(23,27,25,28);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,19,20,16)(21,25,26,23)(22,27,24,28)Polycyclic presentation:
2^9|b^2=fg,c^2=fi,d^2=fi,e^2=i,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 | #72232298, #72232314, #72232318, … |
| maximal normal subgroup of | #72232298, #72232314, #72232318, … |
| maximal quotient of | #72329026, #72335850, C2×#3792571 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3699767);