Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3332 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)(15,18)(16,19)(17,20)(21,22,24,26)(23,27,28,25);(6,8)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23,24,28)(22,25,26,27);(7,12)(10,11)(15,20)(17,18)(22,26)(25,27);(1,2,3,4)(5,7,9,12)(6,10,8,11)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(5,8,9,6)(7,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fh,b^2=g,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72328935, #72328999, #72329056, … |
| maximal normal subgroup of | #72328935, #72328999, #72329056, … |
| maximal quotient of | #72329964, #72333177 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1104698);