Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3816 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)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22,24,26)(23,27,28,25);(5,6,9,7)(8,10,12,11)(13,15)(14,18)(16,19)(17,20)(23,28)(25,27);(1,2)(3,4)(21,23)(22,25)(24,28)(26,27);(5,7,9,6)(8,10,12,11)(13,16,17,18)(14,15,19,20);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=f,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 | #72328203, #72328265, #72328349 |
| maximal normal subgroup of | #72328203, #72328265, #72328349 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1176213);