#201708

order	512
nilpotency class	3
isoclinism	isoclinic class of #201701
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2170 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)(3,5,7,8)(4,6)(9,10)(11,15)(12,22,24,16)(13,25)(14,18)(17,26)(19,27,32,35)(20,33)(21,29)(23,30,37,36)(28,34)(31,39,40,38);(1,3)(2,5,6,8)(4,7)(9,11,14,21)(10,15,18,29)(12,19,24,32)(13,20,26,34)(16,27,22,35)(17,28,25,33)(23,31,37,40)(30,38,36,39);(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,22)(20,31)(21,32)(25,36)(26,37)(28,38)(29,35)(33,39)(34,40);(3,7)(5,8)(9,13,14,26)(10,17,18,25)(11,20,21,34)(12,23,24,37)(15,28,29,33)(16,30,22,36)(19,31,32,40)(27,38,35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi

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, 108904);