#138346

order	512
nilpotency class	4
isoclinism	isoclinic class of #138195
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#32942	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

585 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,5,7,8)(4,6)(9,10,12,17,27,37,39,31,14,19,28,30,13,18,25,15)(11,20,21,29,22,35,38,36,24,34,26,40,23,16,32,33);(1,3,4,7)(2,5)(6,8)(10,15)(12,25)(13,27)(16,29)(17,18)(19,31)(20,33)(21,32)(22,23)(26,38)(28,39)(30,37)(34,36)(35,40);(3,7)(5,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,29)(17,36)(18,34)(19,35)(20,37)(21,28)(22,27)(25,38)(30,33)(31,40)(32,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=ei,f^c=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, 45542);