Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,16)(8,20,23,12)(9,13)(10,15)(11,18)(14,25)(17,31,21,34)(19,30)(22,29,36,28)(24,27)(26,35,33,32);(5,7)(6,11)(8,21,23,17)(9,24)(10,19)(12,26,20,33)(13,16)(14,30)(15,27)(18,25)(22,32,36,35)(28,31,29,34);(1,2,3,4)(5,8)(6,12)(7,17)(9,22)(10,23)(11,26)(13,28)(14,29)(15,20)(16,31)(18,32)(19,21)(24,35)(25,36)(27,33)(30,34);(6,13)(9,25)(11,16)(12,28)(14,15)(18,24)(20,29)(22,36)(26,31)(27,30)(32,35)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237649, #72262411, #72262422, … |
| maximal normal subgroup of | #72237649, #72262411, #72262422, … |
| maximal quotient of | #72314558 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276088);