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×#3488 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11448 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)(5,6,7,8)(9,10,15,17)(11,20,22,16)(12,18,21,14)(13,19,23,24);(5,6)(7,8)(9,11,15,22)(10,16,17,20)(12,13,21,23)(14,19,18,24);(1,3,4,2)(10,17)(14,18)(16,20)(19,24);(9,12)(10,18)(11,13)(14,17)(15,21)(16,24)(19,20)(22,23);(9,13)(10,19)(11,21)(12,22)(14,20)(15,23)(16,18)(17,24);(9,14,15,18)(10,12,17,21)(11,19,22,24)(13,20,23,16)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,f^2=h,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
| maximal subgroup of | #72234479, #72234481, #72234485, … |
| maximal normal subgroup of | #72234479, #72234481, #72234485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533039);