#138309

order	512
nilpotency class	4
isoclinism	isoclinic class of #137944
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	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

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

normal subgroups only

This group has 620 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)(11,20)(12,25,27,17)(13,18)(14,19)(15,22,33,21)(16,26,35,29)(23,37)(24,38)(28,31)(30,34)(32,40,36,39);(2,6,8,7)(9,11,12,21)(10,15,25,38)(13,28,26,40)(14,24,27,22)(16,31,34,32)(17,20,19,33)(18,36,35,37)(23,29,39,30);(1,3,5,4)(2,6,8,7)(9,12)(10,16)(11,21)(13,29)(14,27)(15,31)(17,18)(19,35)(20,36)(22,24)(23,40)(25,34)(26,30)(28,39)(32,38)(33,37)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gh,d^2=ghi,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,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, 45505);