#387095

order	512
nilpotency class	3
isoclinism	isoclinic class of #213808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294291);