#484779

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	#23444	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1097 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)(11,19)(12,23)(13,17)(14,18)(15,21)(16,24)(20,32)(22,33)(25,30)(26,31)(27,39)(28,36)(29,38)(34,40)(35,37);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,18,28)(12,20,24,34)(13,21,26,36)(16,27,23,37)(17,19,31,33)(25,35,38,39)(29,40,30,32);(1,3,5,4)(2,6,7,8)(9,12)(10,16,18,23)(11,20)(13,25)(14,24)(15,27,28,37)(17,30,31,29)(19,32,33,40)(21,35)(22,34)(26,38)(36,39);(12,24)(16,23)(20,34)(25,38)(27,37)(29,30)(32,40)(35,39)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,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, 391975);