Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61105 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13,16,30)(5,21)(6,22)(9,26,28,14)(10,24)(11,12)(15,31,17,23)(18,19)(25,27,29,32);(1,2)(3,9)(4,6)(5,11)(7,12)(8,21)(10,18)(13,27)(14,23)(15,29)(16,28)(17,25)(19,22)(20,24)(26,31)(30,32);(1,3)(2,9)(4,14)(5,15)(6,23)(7,16)(8,25)(10,13)(11,29)(12,28)(17,21)(18,27)(19,32)(20,26)(22,30)(24,31);(1,4)(2,10)(3,14)(5,19)(6,11)(7,20)(8,24)(9,13)(12,22)(15,32)(16,26)(17,27)(18,21)(23,29)(25,31)(28,30)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,f^c=fhi,g^c=gi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72214153, #72214217, … |
| maximal normal subgroup of | #72214039, #72214153, #72214217, … |
| maximal quotient of | #72216438, #72227031, #72227032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404604);