#176497

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

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

normal subgroups only

This group has 679 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,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, 83693);