Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2613 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4201 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)(5,6,9,12)(7,11,8,10)(13,14,17,19)(15,16,20,18)(23,27)(24,28);(5,7,9,8)(6,10,12,11)(13,15,17,20)(14,18,19,16)(21,22,25,26)(23,28,27,24);(7,8)(10,11)(14,19)(15,20)(21,23)(22,24)(25,27)(26,28);(1,2,3,4)(5,8,9,7)(6,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28);(5,6)(7,10)(8,11)(9,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72327161, #72327219, #72327257, … |
| maximal normal subgroup of | #72327161, #72327219, #72327257, … |
| maximal quotient of | #72328382, #72328394 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4364662);