Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14,17,22,19,24,18,23)(15,25,27,20,16,21,26,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,16)(14,20,24,28)(17,26,18,27)(21,22,25,23);(6,9)(10,12)(13,15,19,16)(14,20,24,28)(17,26,18,27)(21,22,25,23);(6,9)(10,12)(13,16,19,15)(14,21,24,25)(17,26,18,27)(20,22,28,23);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,26,16,27)(20,25,28,21)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256959, #72257043, #72257082, … |
| maximal normal subgroup of | #72256959, #72257043, #72257082, … |
| maximal quotient of | #72264228, #72264320, #72308927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398725);