#130587

order	512
nilpotency class	4
isoclinism	isoclinic class of #130522
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2356 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

531 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37783);