#200561

order	512
nilpotency class	3
isoclinism	isoclinic class of #200412
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1358 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)(9,10,14,18,15,19,13,26)(11,16,21,31,22,32,27,30)(12,23,24,33,25,17,28,34)(20,29,37,39,36,38,35,40);(1,2,6,8,4,7,3,5)(9,11,13,21,15,22,14,27)(10,16,18,30,19,32,26,31)(12,20,24,35,25,36,28,37)(17,29,33,39,23,38,34,40);(9,12)(10,17)(11,20)(13,24)(14,28)(15,25)(16,29)(18,33)(19,23)(21,35)(22,36)(26,34)(27,37)(30,39)(31,40)(32,38);(9,13,15,14)(10,18,19,26)(11,21,22,27)(12,24,25,28)(16,30,32,31)(17,33,23,34)(20,35,36,37)(29,39,38,40)

Polycyclic presentation:

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

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, 107757);