Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,13)(5,10)(6,15)(7,11)(8,12)(14,16)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(1,3,7,4)(2,8,11,16)(5,12,6,14)(9,10,13,15);(17,19,21,27,22,18,23,25)(20,26,29,32,28,24,30,31);(1,4,7,3)(2,9,11,13)(5,12,6,14)(8,10,16,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,5,7,6)(2,10,11,15)(3,12,4,14)(8,13,16,9)(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,f^a=fh,f^b=fh,e^d=ehi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72260382, #72287402, #72321346, … |
| maximal normal subgroup of | #72260382, #72287402, #72321346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253067);