Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1889 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,27)(18,28)(19,24)(20,25)(21,26)(22,29)(23,31)(30,35)(32,34)(33,36);(5,7,9,18,10,16,21,30)(6,11,13,23,14,22,26,33)(8,17,20,31,19,29,32,36)(12,15,25,28,24,27,34,35);(1,3,4,2)(7,16)(8,19)(11,22)(15,27)(18,30)(20,32)(23,33)(28,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,g^2=i,b^a=bd,c^a=ce,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311919, #72311922, #72311931, … |
| maximal normal subgroup of | #72311919, #72311922, #72311931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54528);