#7237057

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1394 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7144253);