#134159

order	512
nilpotency class	4
isoclinism	isoclinic class of #133903
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2296 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

727 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 82 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)(11,19,20,34)(12,17)(13,25,26,16)(14,18)(15,21,29,35)(22,32)(23,38,39,30)(24,31)(27,40)(28,37)(33,36);(2,5)(3,7)(9,11)(10,15,18,29)(12,21)(13,22)(14,20)(16,28,25,27)(17,19,31,34)(23,37)(24,35)(26,36)(30,33,38,32)(39,40);(10,16)(11,20)(15,27)(17,30)(18,25)(19,32)(21,35)(22,36)(28,29)(31,38)(33,34)(37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fgi,f^c=fh,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, 41355);