#212477

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
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	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

587 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)(3,8)(4,6)(5,7)(9,10,13,17)(11,14,18,22)(12,15,20,16)(19,21,24,23)(25,26,30,33,29,32,38,28)(27,34,37,36,35,31,39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24)(25,27,29,35)(26,31,32,34)(28,36,33,40)(30,37,38,39);(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(26,32)(27,35)(28,33)(37,39);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(19,24)(21,23)(25,28,38,32,29,33,30,26)(27,36,39,34,35,40,37,31)

Polycyclic presentation:

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

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, 119673);