#143112

order	512
nilpotency class	4
isoclinism	isoclinic class of #143106
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

357 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,22,27)(19,28,29,24)(20,23,34,38)(21,35,33,26)(25,42,37,32)(30,39,45,36)(31,41,46,44)(40,48,43,47);(6,15)(9,16)(18,23)(19,29)(20,32)(21,33)(24,39)(25,26)(27,38)(28,36)(30,47)(34,42)(35,37)(40,44)(41,43)(45,48);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,19,21,31,22,29,33,46)(18,24,26,41,27,28,35,44)(20,30,32,48,34,45,42,47)(23,36,37,43,38,39,25,40)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=cf,d^b=dg,g^b=gi,d^c=dh,e^c=ehi,f^c=fi

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, 50308);