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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3919 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)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,17)(18,20)(21,22,25,27)(23,28,24,26);(6,8)(10,11)(14,17)(18,19)(21,23)(22,26)(24,25)(27,28);(13,15)(14,18)(16,20)(17,19)(21,22)(23,28)(24,26)(25,27);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,20)(17,18)(21,22)(23,26)(24,28)(25,27);(5,8,9,6)(7,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,24,25,23)(22,26,27,28)Polycyclic presentation:
2^9|a^2=fgh,d^2=fg,e^2=fgh,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 | #72232601, #72232612, #72232649, … |
| maximal normal subgroup of | #72232601, #72232612, #72232649, … |
| maximal quotient of | #72328872, #72329174, #72330317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 690931);