#315172

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=efh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^c=fi,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, 222368);