#135046

order	512
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1802
prime radical	C₂³	#474
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

287 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18)(19,27)(20,23)(21,33)(22,36)(24,29)(25,35)(26,32)(28,43)(30,47)(31,44)(34,42)(37,41)(38,45)(39,46)(40,48);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)(18,23)(24,38)(25,39)(26,36)(27,43)(33,44)(40,42)(41,47);(17,19,20,28)(18,24,23,38)(21,34,31,48)(22,37,32,30)(25,41,39,47)(26,42,36,40)(27,44,43,33)(29,46,45,35)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42242);