#134164

order	512
nilpotency class	4
isoclinism	isoclinic class of #133908
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2296 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 668 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)(11,20,23,18)(12,24)(13,17)(14,19)(15,22,30,39)(16,21,32,37)(25,40)(26,35)(27,29)(28,34)(31,36,33,38);(2,5)(3,7)(9,11,14,23)(10,15)(12,21,26,37)(13,22,28,39)(16,29)(17,20)(18,34)(19,30)(24,31)(25,36,27,38)(32,40)(33,35);(9,12,14,26)(10,16,19,32)(11,21,23,37)(13,27,28,25)(15,29,30,40)(17,33,34,31)(18,24,20,35)(22,38,39,36)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,d^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=fgi,d^c=di,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, 41360);