Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4146 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)(7,8)(9,11)(10,12)(13,14,16,18)(15,17,20,19)(22,26)(23,24);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,28)(25,26);(6,11)(7,12)(13,14,16,18)(15,17,20,19)(21,23,25,24)(22,27,26,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,19)(16,18)(17,20)(21,22)(23,28)(24,27)(25,26);(5,8)(6,12)(7,11)(9,10)(13,15,16,20)(14,17,18,19)(21,24,25,23)(22,28,26,27)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72326450, #72326493, #72326502, … |
| maximal normal subgroup of | #72326450, #72326493, #72326502, … |
| maximal quotient of | #72328540, #72328684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1305574);