#145217

order	512
nilpotency class	4
isoclinism	isoclinic class of #145216
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,10)(4,6)(5,9)(7,11)(8,13)(12,15)(14,16)(17,18,22,26)(19,27,28,45)(20,31,32,48)(21,36,34,25)(23,38,39,29)(24,41,42,35)(30,43,37,46)(33,40,47,44);(2,6)(3,11)(4,13)(7,15)(8,9)(10,14)(20,32)(24,42)(27,45)(29,38)(31,48)(33,47)(35,41)(43,46);(1,3,4,12,5,11,13,16)(2,7,8,15,9,10,6,14)(17,19,21,37,22,28,34,30)(18,23,25,44,26,39,36,40)(20,33,35,38,32,47,41,29)(24,43,31,45,42,46,48,27)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52413);