#128637

order	512
nilpotency class	3
isoclinism	isoclinic class of #105688
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	C₂×#41	M₁₆
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgi,c^b=ch,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, 35833);