#6452267

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
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	#23444	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3457 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)(19,27)(20,28)(21,29)(22,34)(23,30)(24,31)(25,37)(26,32)(33,38)(35,40)(36,39);(9,11)(10,15)(12,22,24,19)(13,25,26,20)(14,21)(16,27,31,34)(17,28,32,37)(18,29)(23,33)(30,38)(35,36)(39,40);(1,3,5,4)(2,6,7,8)(9,12,14,24)(10,16,18,31)(11,19,21,22)(13,23,26,36)(15,27,29,34)(17,30,32,39)(20,33,25,35)(28,38,37,40);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,20)(12,23)(14,26)(15,28)(16,30)(18,32)(19,33)(21,25)(22,35)(24,36)(27,38)(29,37)(31,39)(34,40);(10,18)(15,29)(16,31)(17,32)(27,34)(28,37)(30,39)(38,40)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=g,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,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, 6359463);