#244179

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	#2397
prime radical	#2397	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=eh,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,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, 151375);