#263312

order	512
nilpotency class	3
isoclinism	isoclinic class of #263311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

916 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,d^c=dh,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, 170508);