#130169

order	512
nilpotency class	4
isoclinism	isoclinic class of #130114
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

558 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37365);