#6891166

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6798362);