#314834

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

582 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222030);