Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| 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×#2408 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4880 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)(7,11)(8,9)(10,12)(13,14,17,19)(15,16,20,18)(21,22)(23,27)(24,25)(26,28);(6,9)(10,11)(13,15,17,20)(14,18,19,16)(22,24)(26,27);(13,15)(14,16)(17,20)(18,19)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24,25,22)(23,27,28,26);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,18)(19,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72326261, #72326266, #72326282, … |
| maximal normal subgroup of | #72326261, #72326266, #72326282, … |
| maximal quotient of | #72328838, #72329023, #72329855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 524772);