#150351

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 293 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)(9,10,12,17,14,19,25,34)(11,20,22,37,21,35,38,33)(13,16,28,32,29,18,26,36)(15,24,30,27,31,39,40,23);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15,19,31)(12,24)(13,27)(14,21)(16,20,18,35)(17,33,34,37)(22,28)(23,29)(25,39)(26,38)(30,36,40,32);(1,3,5,4)(2,6,7,8)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=ghi,d^2=gi,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dg,e^b=efg,f^b=fi,e^d=ehi,f^e=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, 57547);