#138245

order	512
nilpotency class	4
isoclinism	isoclinic class of #137944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

589 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ghi,d^2=gh,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45441);