Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
3517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12)(15,19)(18,20)(21,22,25,27)(23,28,24,26);(1,2,4,3)(6,9)(10,12)(13,14,17,16)(15,18,19,20)(21,23,25,24)(22,26,27,28);(2,3)(7,11)(10,12)(23,24)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,28)(24,26)(25,27);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,20,19,18)(21,24,25,23)(22,26,27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fghi,e^2=fghi,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 | #72327504, #72327524, #72327581, … |
| maximal normal subgroup of | #72327504, #72327524, #72327581, … |
| maximal quotient of | #72332746 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2410588);