#7222911

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200002
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7130107);