#484670

order	512
nilpotency class	4
isoclinism	isoclinic class of #484587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

951 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391866);