#128516

order	512
nilpotency class	3
isoclinism	isoclinic class of #106017
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂³	#405
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 239 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,7)(3,9,10,6)(5,13,12,8)(11,14,16,15)(17,18,20,25,21,26,31,39,22,23,30,34,32,35,44,46)(19,24,28,37,27,38,40,48,29,33,43,45,41,36,42,47);(1,3,5,11,4,10,12,16)(2,6,8,14,7,9,13,15)(18,23)(19,27,29,41)(20,30)(24,36,33,38)(26,35)(28,42,43,40)(31,44)(37,48,45,47);(17,19,20,28,21,27,31,40,22,29,30,43,32,41,44,42)(18,24,25,37,26,38,39,48,23,33,34,45,35,36,46,47)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,e^2=h,f^2=g,h^2=i,b^a=bd,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, 35712);