#178503

order	512
nilpotency class	3
isoclinism	isoclinic class of #178241
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
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

865 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 865 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85699);