#481688

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1129 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,d^c=di,f^c=fi,f^e=fi

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, 388884);