#481577

order	512
nilpotency class	4
isoclinism	isoclinic class of #481466
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

814 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)(10,16,18,30)(12,25)(13,14)(17,29,32,39)(19,26,20,31)(22,36)(23,24)(27,28)(33,37,34,40)(35,38);(1,2)(3,6)(4,8)(5,7)(9,10,15,18)(11,17,21,32)(12,26,28,31)(13,20,14,19)(16,25,30,27)(22,37,35,40)(23,34,24,33)(29,36,39,38);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,22)(13,23)(14,24)(15,21)(16,29)(18,32)(19,33)(20,34)(25,36)(26,37)(27,38)(28,35)(30,39)(31,40);(11,21)(17,32)(22,35)(23,24)(29,39)(33,34)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388773);