#143072

order	512
nilpotency class	4
isoclinism	isoclinic class of #143009
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×C₄×Q₈	C₂³
prime radical	C₂³	#392
LCS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

433 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,e^c=egi,f^c=fg,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50268);