Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,28)(25,26);(6,8)(10,12)(14,18)(16,19)(21,23)(22,24)(25,28)(26,27);(1,2,4,3)(7,11)(10,12)(14,18)(15,20)(22,26)(23,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,27)(24,26)(25,28);(1,3,4,2)(5,8,9,6)(7,12,11,10)(13,16)(14,15)(17,19)(18,20)(21,24,25,27)(22,23,26,28)Polycyclic presentation:
2^9|a^2=fhi,c^2=ghi,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 | #72333996, #72334003, #72334045 |
| maximal normal subgroup of | #72333996, #72334003, #72334045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1091598);