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 | |
3429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(21,22,25,28)(23,24,27,26);(2,4)(6,10)(8,11)(13,15,16,20)(14,17,18,19)(21,23,25,27)(22,26,28,24);(1,2)(3,4)(6,10)(7,12)(14,18)(17,19)(21,24,25,26)(22,27,28,23);(5,7)(6,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,26)(27,28);(5,8)(6,7)(9,11)(10,12)(13,15)(14,17)(16,20)(18,19)(21,23)(22,24)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fg,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 | #72234728, #72234807, #72234815, … |
| maximal normal subgroup of | #72234728, #72234807, #72234815, … |
| maximal quotient of | #72329870, #72329958, #72332953 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1021650);