Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| 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 | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,17)(12,13)(14,16)(15,20)(18,24)(19,21)(22,23);(1,2,3,4)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,22)(23,24);(10,16)(13,23)(15,17)(19,24);(9,11)(10,17)(12,21)(13,19)(14,20)(15,16)(18,22)(23,24);(9,12)(10,13)(11,18)(14,22)(15,19)(16,23)(17,24)(20,21);(9,13)(10,12)(11,19)(14,23)(15,18)(16,22)(17,21)(20,24)Polycyclic presentation:
2^9|a^2=i,b^2=gh,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
| maximal subgroup of | #72284863, #72285076 |
| maximal normal subgroup of | #72284863, #72285076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532974);