#138395

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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

605 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=ghi,d^2=ghi,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=fhi,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, 45591);