#134088

order	512
nilpotency class	4
isoclinism	isoclinic class of #133885
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

888 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41284);