#244170

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
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	#10012	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

687 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 687 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151366);