#218163

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

549 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)(3,6)(4,5)(7,8)(9,10,13,18)(11,20,22,15)(12,19,14,17)(16,23,24,21)(25,26,29,34,31,28,30,33)(27,32,35,40,37,38,36,39);(2,5)(3,7)(9,11)(10,15)(12,23)(13,22)(14,21)(16,19)(17,24)(18,20)(25,27,30,36,31,37,29,35)(26,32,33,39,28,38,34,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24)(25,28,31,26)(27,32,37,38)(29,33,30,34)(35,40,36,39);(2,5)(6,8)(9,12,13,14)(10,17,18,19)(11,21,22,23)(15,16,20,24)(25,29,31,30)(26,33,28,34)(27,35,37,36)(32,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,d^b=dhi,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, 125359);