#129208

order	512
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1420 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1420 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=di,f^b=fgi,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, 36404);