#142877

order	512
nilpotency class	4
isoclinism	isoclinic class of #142801
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	#121
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

471 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=g,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,f^b=fgi,h^b=hi,d^c=dh,e^c=ei,f^c=fgi,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, 50073);