Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| 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 | C2×#2408 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4596 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)(5,6)(7,10)(8,11)(9,12)(13,14,16,19)(15,18,20,17)(21,22,26,28)(23,24,25,27);(5,7,8,12)(6,9,11,10)(13,15,16,20)(14,17,19,18)(21,23,26,25)(22,27,28,24);(13,14)(15,18)(16,19)(17,20)(21,24)(22,23)(25,28)(26,27);(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,17)(16,19)(18,20)(21,25,26,23)(22,24,28,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,24)(22,25)(23,28)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72232494, #72232503, #72232549, … |
| maximal normal subgroup of | #72232494, #72232503, #72232549, … |
| maximal quotient of | #72328484, #72328850, #72328937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 684692);