#200464

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,g^2=h,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, 107660);