#146118

order	512
nilpotency class	4
isoclinism	isoclinic class of #146118
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	Q₁₆
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

201 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=dh,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fgi,c^b=cf,d^b=dg,e^b=eg,d^c=di,e^c=eh,f^c=fg,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53314);