#6450020

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
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	C₂²×C₄	C₂⁵
prime radical	C₂×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3008 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=g,h^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,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, 6357216);