#142871

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

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

normal subgroups only

This group has 479 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)(21,33)(24,38)(25,41)(26,37)(28,44)(29,34)(30,45)(31,47)(35,39)(36,43)(40,48)(42,46);(1,3,5,13)(2,7,9,16)(4,14,6,12)(8,11,15,10)(17,18)(19,24)(20,23)(21,25)(22,26)(27,39)(28,35)(29,40)(30,46)(31,36)(32,37)(33,43)(34,42)(38,44)(41,47)(45,48);(1,2)(3,11)(4,8)(5,9)(6,15)(7,12)(10,13)(14,16)(17,19,22,29)(18,24,26,40)(20,30,32,28)(21,34,33,27)(23,35,37,46)(25,42,43,39)(31,44,47,45)(36,48,41,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,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, 50067);