#201757

order	512
nilpotency class	3
isoclinism	isoclinic class of #201678
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	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1944 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)(3,7,8,5)(4,6)(9,10,13,18,14,17,19,31)(11,15,22,27,21,28,25,32)(12,16,24,30,23,29,33,39)(20,26,35,38,34,37,36,40);(1,2)(3,5)(4,6)(7,8)(9,11,19,22,14,21,13,25)(10,15,18,32,17,28,31,27)(12,20,33,35,23,34,24,36)(16,26,30,40,29,37,39,38);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,20)(13,24)(14,23)(15,26)(17,29)(18,30)(19,33)(21,34)(22,35)(25,36)(27,38)(28,37)(31,39)(32,40);(12,23)(16,29)(20,34)(24,33)(26,37)(30,39)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,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, 108953);