#6444378

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351574);