#217433

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

562 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)(4,6)(5,8)(11,17)(12,19)(14,20)(16,22)(18,23)(21,24)(25,26,30,33,28,31,29,38)(27,34,37,39,35,40,36,32);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,14,18,21)(12,16,19,22)(17,20,23,24)(26,31)(27,35)(33,38)(36,37);(3,8)(5,7)(9,11,12,17,13,18,19,23)(10,14,16,20,15,21,22,24)(25,27,29,36,28,35,30,37)(26,32,33,34,31,39,38,40);(2,6)(5,7)(10,15)(11,18)(16,22)(17,23)(26,31)(32,39)(33,38)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124629);