#6447726

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3036 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)(3,7,8,6)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,33)(23,32)(25,31)(26,39)(27,34)(28,36)(29,35)(30,37)(38,40);(2,5)(6,7)(9,11,13,24,15,25,14,23)(10,16,19,21,20,32,18,31)(12,26,27,22,28,37,29,38)(17,33,35,39,36,40,34,30);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,35)(20,36)(21,33)(23,26)(24,37)(25,38)(31,40)(32,39);(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,35)(20,36)(21,33)(23,26)(24,37)(25,38)(31,40)(32,39);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,21,32,31)(17,34,36,35)(22,26,38,37)(30,33,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6354922);