#243923

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

715 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

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, 151119);