#332843

order	512
nilpotency class	3
isoclinism	isoclinic class of #332839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 770 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,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, 240039);