Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3528 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)(5,6,10,11)(7,12,8,9)(13,14,17,19)(15,20,16,18)(22,26)(24,28);(6,11)(7,8)(13,14)(15,18)(16,20)(17,19)(22,26)(23,27);(5,7,10,8)(6,9,11,12)(13,15,17,16)(14,18,19,20)(21,22,25,26)(23,24,27,28);(1,2,3,4)(5,8,10,7)(6,9,11,12)(13,16,17,15)(14,18,19,20)(21,23,25,27)(22,24,26,28);(5,9)(6,7)(8,11)(10,12)(13,14)(15,20)(16,18)(17,19)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72327118, #72327159, #72327169 |
| maximal normal subgroup of | #72327118, #72327159, #72327169 |
| maximal quotient of | #72330438 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4571821);