#200571

order	512
nilpotency class	3
isoclinism	isoclinic class of #200418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1297 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)(9,10,13,18,14,17,19,34)(11,15,22,31,21,30,27,35)(12,23,26,36,25,16,28,33)(20,29,37,39,24,32,38,40);(1,2,6,8,4,7,3,5)(9,11,19,22,14,21,13,27)(10,15,18,35,17,30,34,31)(12,24,26,37,25,20,28,38)(16,32,36,40,23,29,33,39);(2,7)(5,8)(9,12,14,25)(10,16,17,23)(11,20,21,24)(13,28,19,26)(15,29,30,32)(18,36,34,33)(22,38,27,37)(31,40,35,39);(10,17)(15,30)(16,23)(18,34)(29,32)(31,35)(33,36)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,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, 107767);