#6917307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884809
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^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=dfh,g^c=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6824503);