#200594

order	512
nilpotency class	3
isoclinism	isoclinic class of #200424
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1137 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)(9,10,14,18,15,19,13,27)(11,16,23,30,22,31,21,34)(12,24,25,32,26,17,28,33)(20,29,35,40,36,38,37,39);(1,2,3,5,4,6,7,8)(9,11,14,21,15,22,13,23)(10,16,27,30,19,31,18,34)(12,20,28,35,26,36,25,37)(17,29,33,39,24,38,32,40);(9,12,15,26)(10,17,19,24)(11,20,22,36)(13,25,14,28)(16,29,31,38)(18,32,27,33)(21,35,23,37)(30,39,34,40);(2,6)(5,8)(9,13,15,14)(10,18,19,27)(11,21,22,23)(12,25,26,28)(16,30,31,34)(17,32,24,33)(20,35,36,37)(29,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,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, 107790);