#128578

order	512
nilpotency class	3
isoclinism	isoclinic class of #106004
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1749
prime radical	C₂³	#407
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

354 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35774);