Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161262 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3506 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13,15,10)(8,16)(9,12)(11,17)(14,18,20,19)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(2,4)(8,17)(11,16)(14,20)(18,19)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,29)(27,36)(28,33)(30,35)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239686 |
| maximal normal subgroup of | #72239686 |
| maximal quotient of | #72239736 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92296);