Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×#97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6,8,11,9,7,10,12)(13,14,18,24,17,23,22,25,21,15,27,28,20,26,19,16);(2,4)(13,15,19,16,20,23,27,28,21,14,22,25,17,26,18,24);(5,7,8,12,9,6,10,11)(13,16,22,23,20,28,18,14,21,25,19,26,17,24,27,15)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ci,f^a=fi,c^b=ch,e^b=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225528, D4×#1313, #72225546, … |
| maximal normal subgroup of | #72225528, D4×#1313, #72225546, … |
| maximal quotient of | #72225527, #72225528, C2×#129062, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36258);