Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,24,20,22);(2,5)(6,8)(17,19,21,20)(18,22,23,24);(10,13)(14,16)(19,20)(22,24);(1,3,4,7)(2,6,5,8)(17,20,21,19)(18,22,23,24);(19,20)(22,24);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di,e^d=ei140 conjugacy classes — view the full character table.
| maximal subgroup of | #72304318, #72304335, #72304336, … |
| maximal normal subgroup of | #72304318, #72304335, #72304336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537295);