#179438

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

853 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,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, 86634);