#150575

order	512
nilpotency class	5
isoclinism	isoclinic class of #150567
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

600 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 60 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)(4,6)(7,8)(9,10,13,17,14,19,29,36)(11,20,22,34,23,31,38,39)(12,18,25,35,26,16,28,33)(15,27,30,24,32,37,40,21);(2,5)(3,8)(6,7)(9,11,14,23)(10,15,19,32)(12,24,26,21)(13,27,29,37)(16,31,18,20)(17,34,36,39)(22,28,38,25)(30,35,40,33);(2,6)(5,7)(9,12,14,26)(10,16,19,18)(11,21,23,24)(13,28,29,25)(15,20,32,31)(17,35,36,33)(22,37,38,27)(30,39,40,34)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,e^c=ei,e^d=ehi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57771);