Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| 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 | C2×#2537 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,20)(17,18)(23,27)(26,28);(6,8)(10,11)(14,18)(16,19)(22,25)(26,28);(7,12)(10,11)(14,18)(15,20)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,20)(14,19,18,16)(21,23,24,27)(22,26,25,28);(5,8,9,6)(7,11,12,10)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^9|a^2=fg,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72297810, #72297825, #72297837, … |
| maximal normal subgroup of | #72297810, #72297825, #72297837, … |
| maximal quotient of | #72298386 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 596277);