Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | |
3811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,7)(10,11)(15,19)(18,20)(22,26)(23,24);(5,6,9,7)(8,10,12,11)(13,14,17,16)(15,18,19,20)(21,22)(23,28)(24,27)(25,26);(1,2,4,3)(21,23,25,24)(22,27,26,28);(1,2,4,3)(5,7,9,6)(8,10,12,11)(13,15)(14,18)(16,20)(17,19)(21,24,25,23)(22,27,26,28);(1,3,4,2)(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,20,19,18)(21,22)(23,27)(24,28)(25,26)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232385, #72232401, #72232402, … |
| maximal normal subgroup of | #72232385, #72232401, #72232402, … |
| maximal quotient of | #72278797, C2×#1368921, #72299194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1276117);