#130484

order	512
nilpotency class	4
isoclinism	isoclinic class of #130482
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×C₄×D₄	C₂³
prime radical	#45269	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 750 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,e^c=ei,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, 37680);