#6441700

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2110 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)(3,5)(4,6)(7,8)(9,10,15,18,14,20,13,19)(11,21,23,16,25,36,24,31)(12,17,29,33,28,35,27,34)(22,37,38,30,26,40,39,32);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,26)(13,23)(14,25)(15,24)(17,32)(18,21)(19,36)(20,31)(22,28)(27,39)(29,38)(30,35)(33,40)(34,37);(2,6)(5,8)(9,12,13,27,14,28,15,29)(10,17,19,34,20,35,18,33)(11,22,24,39,25,26,23,38)(16,30,21,37,31,32,36,40);(9,12,13,27,14,28,15,29)(10,17,19,34,20,35,18,33)(11,22,24,39,25,26,23,38)(16,30,21,37,31,32,36,40);(9,13,14,15)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,21,31,36)(17,33,35,34)(22,38,26,39)(30,37,32,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^c=hi,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, 6348896);