#178010

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46016	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

801 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 128 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85206);