#145435

order	512
nilpotency class	4
isoclinism	isoclinic class of #145372
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52631);