#129886

order	512
nilpotency class	4
isoclinism	isoclinic class of #129781
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

758 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 96 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)(11,20,23,33)(12,17,24,15)(13,18)(14,19)(16,22,31,26)(21,36)(25,32,39,29)(27,34)(28,38)(30,40)(35,37);(1,2)(3,6)(4,8)(5,7)(10,15)(12,24)(16,28)(17,19)(18,29)(20,35)(21,37)(25,39)(30,31)(32,34)(33,36)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,21)(13,26)(14,23)(15,28)(17,30)(18,33)(19,31)(20,34)(22,27)(24,37)(25,40)(29,36)(32,35)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37082);