#135646

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1831
prime radical	C₂³	#423
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

256 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=d,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^b=fi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42842);