#129104

order	512
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#106
prime radical	C₂³	#427
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

256 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

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

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, 36300);