#484388

order	512
nilpotency class	4
isoclinism	isoclinic class of #484176
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×C₄⋊C₄	C₂⁴
prime radical	#18292	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

850 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)(3,4)(7,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,27)(19,34)(20,29)(21,32)(23,33)(26,31)(28,36)(30,37)(35,39)(38,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,32)(18,29)(22,39)(23,35)(24,36)(25,30)(26,38)(27,37)(31,40)(33,34);(1,3,5,4)(2,6,7,8)(9,12,14,23)(10,16,18,31)(11,19,20,35)(13,26,27,24)(15,28,29,40)(17,33,25,22)(21,38,37,36)(30,39,32,34);(11,20)(15,29)(19,35)(21,37)(28,40)(30,32)(34,39)(36,38)

Polycyclic presentation:

2^9|c^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,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, 391584);