#178157

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46089	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1066 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1066 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) = 128

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

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

Polycyclic presentation:

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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85353);