#142898

order	512
nilpotency class	4
isoclinism	isoclinic class of #142798
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

468 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,f^b=fg,h^b=hi,d^c=dh,e^c=ei,f^c=fg,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50094);