#179229

order	512
nilpotency class	3
isoclinism	isoclinic class of #179229
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1135 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 1135 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86425);