#128341

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₄×#99
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

422 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ci,f^a=fgi,d^b=dg,e^b=ei,g^b=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35537);