#128468

order	512
nilpotency class	3
isoclinism	isoclinic class of #106017
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1752
prime radical	#525	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fg,c^b=ch,d^b=dg,e^b=ei,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, 35664);