Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,20)(13,26)(18,33)(19,29)(21,28)(23,36)(24,32)(25,31)(27,35)(30,34);(5,7,10,20,11,21,9,19)(6,12,14,28,16,29,15,17)(8,23,26,18,24,34,25,35)(13,30,32,27,31,36,22,33);(5,8,11,24)(6,13)(7,18)(9,25,10,26)(12,27,29,33)(14,32)(15,22)(16,31)(17,30,28,36)(19,23)(20,34)(21,35);(7,19,21,20)(8,24)(12,28,29,17)(13,31)(18,34,35,23)(22,32)(25,26)(27,30,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=fg,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279537, #72279655, #72279719, … |
| maximal normal subgroup of | #72279537, #72279655, #72279719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294587);