#138314

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45510);