Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499269 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23573 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,12)(4,17)(5,6)(8,22)(9,25)(10,21)(13,30)(14,18)(15,28)(16,29)(19,20)(23,26)(24,31)(27,32);(1,2)(3,13)(4,9)(5,21)(6,10)(7,11)(8,14)(12,26)(15,23)(16,24)(17,25)(18,31)(19,27)(20,32)(22,29)(28,30);(1,3)(2,8)(4,18)(5,16)(6,14)(7,15)(9,26)(10,23)(11,24)(12,17)(13,21)(19,29)(20,28)(22,25)(27,30)(31,32);(3,14,15,16)(4,19)(8,23,24,13)(9,27)(12,18,28,29)(17,20)(22,26,31,30)(25,32)Polycyclic presentation:
2^9|d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72213945, #72214675, … |
| maximal normal subgroup of | #72213937, #72213945, #72214675, … |
| maximal quotient of | #72224865, C2×#499269, #72231546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406465);