Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,27)(24,29)(25,28)(26,31)(30,33)(32,36)(34,35);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,26,32)(22,27,31,36)(24,33,34,25)(28,29,30,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,28)(23,32)(25,33)(27,35)(29,36)(30,31);(6,12)(10,13)(11,20)(18,19)(21,24)(22,29)(23,33)(25,32)(26,34)(27,30)(28,36)(31,35);(1,3,4,2)(23,32)(24,34)(27,36)(29,35)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cfi,e^b=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72280361, #72280484, #72280604, … |
| maximal normal subgroup of | #72280361, #72280484, #72280604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285178);