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×#3498 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
12108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2,4,3)(6,8)(9,10,13,12)(11,14,15,16);(5,6,7,8);(1,3,4,2)(10,12)(14,16);(11,15)(14,16);(9,11)(10,14)(12,16)(13,15);(9,12,13,10)(11,16,15,14)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,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 | #72234475, #72234481, #72262148, … |
| maximal normal subgroup of | #72234475, #72234481, #72262148, … |
| maximal quotient of | #72262124, D4×#2366, #72297627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533050);