#171878

order	512
nilpotency class	3
isoclinism	isoclinic class of #171725
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

784 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 148 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)(9,10,14,17)(11,18,21,23)(12,16,13,22)(15,20,24,19)(30,37)(38,39);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,24)(18,22)(25,26,29,32)(27,30,34,37)(28,35,36,31)(33,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,24,18)(25,27)(26,30)(28,33)(29,34)(31,38)(32,37)(35,39)(36,40);(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,18,24,23)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,38,37,39)

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79074);