#141737

order	512
nilpotency class	4
isoclinism	isoclinic class of #141698
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33717	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

396 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,22,28,38)(14,23)(15,32)(16,20)(17,27,34,36)(18,21)(19,30)(24,31)(25,29,33,39)(26,35,37,40);(1,3,7,2,5,8,4,6)(9,11,21,19,15,23,31,20,16,24,14,32,30,18,13,10)(12,25,27,17,29,38,26,35,28,33,36,34,39,22,37,40);(2,6)(3,8)(9,12,21,36,15,29,31,37,16,28,14,27,30,39,13,26)(10,17,24,38,19,35,18,33,20,34,11,22,32,40,23,25)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egh,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48933);