#146123

order	512
nilpotency class	4
isoclinism	isoclinic class of #146112
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2075
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#480
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53319);