#507739

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=gh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=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, 414935);