Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| 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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,8)(7,11)(13,14,17,16)(15,18,19,20)(21,22)(23,27)(24,28)(25,26);(1,2,3,4)(5,6)(7,12)(8,9)(10,11)(13,14)(15,18)(16,17)(19,20)(22,26)(24,27);(1,2)(3,4)(6,8)(10,12)(14,16)(18,20)(21,23,25,28)(22,27,26,24);(5,7,9,11)(6,10,8,12)(13,15,17,19)(14,18,16,20)(21,22)(23,27)(24,28)(25,26);(5,8,9,6)(7,10,11,12)(13,16,17,14)(15,18,19,20)(21,24,25,27)(22,23,26,28)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=fgi,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232373, #72232400, #72232419, … |
| maximal normal subgroup of | #72232373, #72232400, #72232419, … |
| maximal quotient of | #72323621, #72328853, #72329918 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3936667);