Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500250 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24058 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61630 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
896 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 896 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,7)(5,10)(6,11)(8,15)(9,16)(13,25)(14,29)(17,21)(18,23)(19,26)(20,30)(22,32)(24,31)(27,28);(2,7)(3,13)(4,16)(8,20)(9,11)(10,21)(12,27)(17,31)(22,23)(24,26)(25,29)(30,32);(1,3,4,14,6,13,16,28)(2,8,9,22,11,23,7,20)(5,18,17,32,19,15,31,30)(10,25,24,29,26,12,21,27);(2,7)(3,13)(4,16)(8,22)(9,11)(10,21)(12,27)(15,18)(17,31)(20,23)(24,26)(25,29)Polycyclic presentation:
2^9|c^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,d^c=dh,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407446);