#6452032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359228);