#6724572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6631768);