Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2812 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3620 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)(7,12)(8,10)(9,11)(14,18)(16,19)(21,22)(23,26)(24,25)(27,28);(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(22,25)(26,27);(1,2,4,3)(5,8,9,12)(6,10,11,7)(13,15,17,20)(14,19,18,16)(21,22)(23,27)(24,25)(26,28);(1,3,4,2)(5,7)(6,8)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20);(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|c^2=i,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213467, #72333438, #72333440, … |
| maximal normal subgroup of | #72213467, #72333438, #72333440, … |
| maximal quotient of | #72267000, #72267503, #72267612 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4323555);