#320274

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

687 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=i,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227470);