#199375

order	512
nilpotency class	3
isoclinism	isoclinic class of #199212
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106571);