#100822

order	512
nilpotency class	3
isoclinism	isoclinic class of #100723
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 347 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=hi,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,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, 8018);