Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4828 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)(6,7)(8,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,27)(24,28)(25,26);(1,2)(3,4)(5,6,9,7)(8,10,12,11)(14,18)(16,19)(22,26)(23,28);(13,15)(14,19)(16,18)(17,20)(21,23)(22,24)(25,28)(26,27);(5,7,9,6)(8,11,12,10)(13,16,17,19)(14,15,18,20)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15)(14,19)(16,18)(17,20)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|b^2=i,d^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72327861, #72327864, #72327866, … |
| maximal normal subgroup of | #72327861, #72327864, #72327866, … |
| maximal quotient of | #72328367, #72328631, #72330324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 787789);