#128340

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#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×C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 436 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:

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

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,f^a=fgi,d^b=dg,e^b=ei,g^b=gi,f^d=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, 35536);