#129786

order	512
nilpotency class	4
isoclinism	isoclinic class of #129781
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	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

909 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 909 conjugacy classes of subgroups — too many to draw. Only its 108 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)(3,5)(4,7)(6,8)(9,10,14,19,15,20,13,18)(11,21,24,17,25,32,23,33)(12,26,27,34,28,16,29,31)(22,36,37,30,38,35,39,40);(2,5)(3,8)(6,7)(10,16)(17,30)(18,31)(19,34)(20,26)(21,35)(32,36)(33,40);(9,11,15,25)(10,17,20,33)(12,22,28,38)(13,24,14,23)(16,30,26,40)(18,32,19,21)(27,39,29,37)(31,36,34,35)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^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, 36982);