#150446

order	512
nilpotency class	5
isoclinism	isoclinic class of #150441
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	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

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

normal subgroups only

This group has 351 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57642);