#145228

order	512
nilpotency class	4
isoclinism	isoclinic class of #145213
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	#3147	C₂²
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

271 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=dh,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,d^c=di,e^c=eghi,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, 52424);