#176249

order	512
nilpotency class	3
isoclinism	isoclinic class of #176173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

762 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,28,30,31)(33,35,37,43)(34,38,40,46)(36,42,44,48)(39,45,41,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)(34,39)(36,44)(38,45)(40,41)(42,48)(46,47);(2,9)(7,16)(8,13)(10,15)(18,24)(22,29)(23,30)(28,31)(34,40)(38,46)(39,41)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83445);