#6452278

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	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,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, 6359474);