Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2807 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 2807 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(11,24)(15,29)(25,36)(31,32);(5,6)(7,15,17,29)(8,12)(9,14)(10,13)(11,18,24,21)(16,32,33,31)(19,27)(20,26)(22,30)(23,28)(25,34,36,35);(5,7)(6,11)(8,16)(9,21)(10,17)(12,25)(13,24)(14,29)(15,28)(18,23)(19,35)(20,33)(22,34)(26,36)(27,31)(30,32);(2,4)(5,8,10,20)(6,12,13,26)(7,16,17,33)(9,22,23,19)(11,25,24,36)(14,30,28,27)(15,31,29,32)(18,35,21,34)Polycyclic presentation:
2^9|b^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72312580, #72312585, #72312596 |
| maximal normal subgroup of | #72312580, #72312585, #72312596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325016);