#176498

order	512
nilpotency class	3
isoclinism	isoclinic class of #176333
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 651 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,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, 83694);