#128639

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	C₂×#41	M₁₆
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 294 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)(19,29,32,40)(20,26)(21,27)(22,28)(23,25)(24,31,37,30)(33,43,46,48)(34,41)(35,39)(36,42)(38,45,47,44);(2,7,10,9)(8,14,16,15)(17,19,21,33,22,31,36,45,23,32,35,46,20,30,34,44)(18,24,27,38,28,29,42,43,25,37,39,47,26,40,41,48);(1,3,5,12,6,13,4,11)(2,8,9,15,10,16,7,14)(18,25)(19,30,32,31)(21,35)(24,29,37,40)(26,28)(33,45,46,44)(34,36)(38,48,47,43)

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=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, 35835);