#6452273

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	#23443	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3315 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)(11,15)(12,16)(13,17)(14,18)(19,22)(20,28)(21,29)(23,25)(24,30)(26,31)(27,32)(33,35)(34,39)(36,38)(37,40);(9,10)(11,19,22,15)(12,23,25,16)(13,17)(14,18)(20,28)(21,31,26,29)(24,36,38,30)(27,32)(33,35)(34,39)(37,40);(1,3,5,4)(2,6,7,8)(9,11,14,22)(10,15,18,19)(12,20,25,35)(13,26,27,21)(16,28,23,33)(17,31,32,29)(24,37,38,34)(30,40,36,39);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,33)(21,34)(22,35)(26,37)(27,38)(29,39)(31,40)(32,36);(9,13,14,27)(10,17,18,32)(11,21,22,26)(12,24,25,38)(15,29,19,31)(16,30,23,36)(20,34,35,37)(28,39,33,40)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=g,e^2=i,b^a=bf,c^a=cg,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, 6359469);