#6752363

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682564
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1309 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei

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, 6659559);