#7237080

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234421
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	#23932	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 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)(25,26,30,33)(27,31,29,38)(28,35,37,39)(32,36,40,34);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,29)(26,31,33,38)(28,36,37,34)(32,35,40,39);(1,3)(2,5)(4,8)(6,7)(25,28)(26,32)(27,34)(29,36)(30,37)(31,39)(33,40)(35,38);(3,8)(5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,27,30,29)(26,31,33,38)(28,34,37,36)(32,39,40,35);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(25,27,30,29)(26,31,33,38)(28,34,37,36)(32,39,40,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=fi,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=ehi,d^c=dhi,g^c=gi

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, 7144276);