Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3381 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)(8,12)(13,14,18,19)(15,16,20,17)(21,22,25,24)(23,27,28,26);(5,6,9,7)(8,11,12,10)(13,15,18,20)(14,17,19,16)(22,24)(26,27);(5,6,9,7)(8,10,12,11)(13,16,18,17)(14,15,19,20)(21,22,25,24)(23,26,28,27);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,17,18,16)(14,15,19,20)(21,23)(22,26)(24,27)(25,28);(5,8)(6,10)(7,11)(9,12)(13,15)(14,17)(16,19)(18,20)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232549, #72232550, #72232571, … |
| maximal normal subgroup of | #72232549, #72232550, #72232571, … |
| maximal quotient of | #72330776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1327305);