#151978

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₄⋊C₁₆
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

240 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^b=di,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59174);