#218572

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	C₂×C₄×D₄
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,d^b=dh,d^c=di,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125768);