#173645

order	512
nilpotency class	3
isoclinism	isoclinic class of #173481
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

831 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 831 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 80841);