#200570

order	512
nilpotency class	3
isoclinism	isoclinic class of #200418
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

1115 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)(9,10,13,18,14,17,28,29)(11,15,21,32,20,31,26,35)(12,22,25,36,24,16,27,34)(19,30,37,39,23,33,38,40);(1,2,3,5,4,6,7,8)(9,11,13,26,14,20,28,21)(10,15,29,32,17,31,18,35)(12,23,27,38,24,19,25,37)(16,33,34,39,22,30,36,40);(2,6)(5,8)(9,12,14,24)(10,16,17,22)(11,19,20,23)(13,27,28,25)(15,30,31,33)(18,36,29,34)(21,38,26,37)(32,40,35,39);(10,17)(15,31)(16,22)(18,29)(30,33)(32,35)(34,36)(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,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, 107766);