#319340

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

752 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 182 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)(9,10,12,15,13,16,21,24)(11,17,18,22,19,23,20,14)(26,30)(27,33)(32,39)(35,36);(1,3,4,8)(2,5,6,7)(25,26,28,32,29,30,37,39)(27,34,36,38,33,40,35,31);(9,11,13,19)(10,14,16,22)(12,20,21,18)(15,23,24,17)(25,27)(26,31)(28,35)(29,33)(30,38)(32,40)(34,39)(36,37);(2,6)(5,7)(25,28,29,37)(26,32,30,39)(27,35,33,36)(31,40,38,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,c^2=h,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226536);