#251432

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46231	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=h,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

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, 158628);