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 | C2×D4×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11292 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)(11,18)(12,15)(13,16)(14,17)(19,23)(20,21)(22,24);(5,6)(7,8)(9,11,14,15)(10,12,17,18)(13,19,22,21)(16,20,24,23);(9,11,14,15)(10,12,17,18)(13,21,22,19)(16,23,24,20);(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24);(9,12)(10,15)(11,17)(13,20)(14,18)(16,21)(19,24)(22,23);(1,3,4,2)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,f^2=i,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
| maximal subgroup of | #72284172, #72284234 |
| maximal normal subgroup of | #72284172, #72284234 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533087);