Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61433 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,9,7,3,5,11)(4,10,13,16,14,8,12,15)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(1,3,7,2)(4,10,14,8)(5,9,6,11)(12,15,13,16);(1,4,6,13,7,14,5,12)(2,8,9,15,3,10,11,16)(17,19,21,27,22,18,23,25)(20,26,29,32,28,24,30,31);(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fh,e^d=ehi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72285760, #72287666, #72287710, … |
| maximal normal subgroup of | #72285760, #72287666, #72287710, … |
| maximal quotient of | #72286059, #72286991, #72343624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253059);