#512769

order	512
nilpotency class	5
isoclinism	isoclinic class of #33982
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,4,3)(5,6)(7,14)(8,18)(9,27)(10,33)(11,15)(12,19)(13,21)(16,30)(17,29)(20,43)(22,52)(23,42)(24,39)(25,56)(26,48)(28,40)(31,59)(32,60)(34,41)(35,45)(36,54)(37,57)(38,51)(44,62)(46,58)(47,63)(49,50)(53,68)(55,67)(61,66)(64,65);(5,7)(6,13)(8,20)(9,28)(10,34)(11,24)(12,23)(14,27)(15,36)(16,41)(17,40)(18,49)(19,38)(21,30)(22,29)(25,57)(26,53)(31,64)(32,61)(33,37)(35,55)(39,59)(42,60)(43,56)(44,52)(45,65)(46,68)(47,67)(48,66)(50,58)(51,63)(54,62);(1,3,4,2)(5,7)(6,14)(8,24)(9,21)(10,22)(11,20)(12,23)(13,27)(15,43)(16,40)(17,41)(18,39)(19,42)(25,54)(26,55)(28,30)(29,34)(31,50)(32,51)(33,52)(35,53)(36,56)(37,44)(38,60)(45,68)(46,65)(47,66)(48,67)(49,59)(57,62)(58,64)(61,63);(5,8,12,11)(6,15,19,18)(7,20,23,24)(9,25,32,31)(10,26,35,29)(13,36,38,49)(14,39,42,43)(16,44,47,46)(17,45,48,33)(21,50,51,54)(22,34,53,55)(27,59,60,56)(28,57,61,64)(30,58,63,62)(37,40,65,66)(41,52,67,68)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419965);