#137713

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

427 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dgi,e^b=ei,f^b=fgi,d^c=di,f^c=fi

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, 44909);