#6452276

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437999
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#62855	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3426 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)(9,10,14,18)(11,15,21,30)(12,16,24,32)(13,25,27,17)(19,28,22,38)(20,35,37,29)(23,33,26,31)(34,40,36,39);(9,11)(10,15)(12,22,24,19)(13,20)(14,21)(16,28,32,38)(17,29)(18,30)(23,36,26,34)(25,35)(27,37)(31,39,33,40);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,19)(13,26)(14,24)(15,28)(17,33)(18,32)(20,36)(21,22)(23,27)(25,31)(29,40)(30,38)(34,37)(35,39);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,28)(17,31)(18,32)(20,34)(21,22)(25,33)(26,27)(29,39)(30,38)(35,40)(36,37);(9,13,14,27)(10,17,18,25)(11,20,21,37)(12,23,24,26)(15,29,30,35)(16,31,32,33)(19,34,22,36)(28,39,38,40)

Polycyclic presentation:

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

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, 6359472);