#174178

order	512
nilpotency class	3
isoclinism	isoclinic class of #174093
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	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

739 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 739 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,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,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81374);