#145218

order	512
nilpotency class	4
isoclinism	isoclinic class of #145218
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	#19189	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 278 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:

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

Polycyclic presentation:

2^9|c^2=dh,d^2=g,e^2=hi,f^2=i,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,f^b=fi,d^c=di,e^c=eghi,f^c=fgi,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52414);