#136148

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

405 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,4,3)(6,10)(7,8)(11,12)(13,14)(15,27)(16,21)(17,23)(18,25)(19,24)(20,26)(22,29)(28,41)(30,36)(31,39)(32,43)(33,37)(34,38)(35,40)(42,44);(1,3,4,2)(5,6)(7,12)(8,10)(9,11)(14,21)(16,17)(18,19)(22,36)(23,24)(25,37)(26,38)(27,41)(28,44)(29,35)(30,31)(33,34)(39,42)(40,43);(5,6)(7,10)(8,12)(9,11)(13,15,20,32)(14,22,26,42)(16,29,34,44)(17,35,33,28)(18,31,19,30)(21,36,38,39)(23,27,37,43)(24,41,25,40)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=gi,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=egi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43344);