Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,11,13,28)(8,19)(9,23)(12,27,30,34)(14,20,24,29)(15,31,33,32)(16,35)(17,25)(21,36)(22,26);(5,6)(7,15)(8,20)(9,24)(10,13)(11,26)(12,16)(14,23)(17,27)(18,33)(19,28)(21,31)(22,29)(25,34)(30,36)(32,35);(1,2,3,4);(5,7,10,18)(6,12,13,30)(8,21,22,35)(9,25,23,17)(11,27,28,34)(14,32,24,31)(15,29,33,20)(16,19,36,26)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286595, #72286839, D4×#2180, … |
| maximal normal subgroup of | #72286595, #72286839, D4×#2180, … |
| maximal quotient of | #72287314, #72287506, #72309545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405061);