#200556

order	512
nilpotency class	3
isoclinism	isoclinic class of #200412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1179 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)(3,7)(6,8)(9,10,13,18,14,17,27,28)(11,15,21,31,20,30,25,33)(12,22,24,34,23,16,26,32)(19,29,36,39,35,38,37,40);(1,2,3,5,4,6,7,8)(9,11,13,25,14,20,27,21)(10,15,28,31,17,30,18,33)(12,19,26,36,23,35,24,37)(16,29,32,40,22,38,34,39);(9,12,14,23)(10,16,17,22)(11,19,20,35)(13,26,27,24)(15,29,30,38)(18,34,28,32)(21,37,25,36)(31,40,33,39);(10,17)(15,30)(16,22)(18,28)(29,38)(31,33)(32,34)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^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, 107752);