#6443181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4956 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)(6,8)(9,10)(11,21)(12,18)(13,20)(14,17)(15,19)(16,22)(23,34)(24,30)(25,32)(26,31)(27,36)(28,33)(29,35)(37,39)(38,40);(1,2,3,4)(9,11,13,22,14,24,15,25)(10,16,19,21,17,32,20,30)(12,26,27,23,28,38,29,37)(18,34,35,31,33,39,36,40);(5,6,7,8)(10,17)(11,22)(13,15)(16,30)(18,33)(21,32)(23,26)(24,25)(27,29)(31,39)(34,40)(37,38);(9,12,14,28)(10,18,17,33)(11,23,24,37)(13,29,15,27)(16,31,32,40)(19,36,20,35)(21,34,30,39)(22,26,25,38);(10,17)(11,24)(18,33)(19,20)(22,25)(23,37)(26,38)(35,36)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350377);