#100689

order	512
nilpotency class	3
isoclinism	isoclinic class of #100555
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

418 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=gh,e^2=gi,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=egi,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7885);