#138280

order	512
nilpotency class	4
isoclinism	isoclinic class of #137944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

730 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45476);