Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3498 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,14,15)(11,16,18,12)(13,21,22,17)(19,20,24,23);(5,6,7,8)(10,15)(12,16)(17,21)(20,23);(10,15)(11,18)(17,21)(19,24);(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,21)(22,24);(1,3,4,2)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24);(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)Polycyclic presentation:
2^9|a^2=i,b^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
| maximal subgroup of | #72238201, #72238208, #72238213, … |
| maximal normal subgroup of | #72238201, #72238208, #72238213, … |
| maximal quotient of | #72263710, D4×#2328, #72297185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532961);