#150539

order	512
nilpotency class	5
isoclinism	isoclinic class of #150443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

287 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=gh,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cfhi,d^b=dg,e^b=efghi,d^c=dhi,f^c=fi,e^d=ehi,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, 57735);