#138315

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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 516 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,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, 45511);