#150267

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dg,e^b=efghi,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, 57463);