#498097

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=deh,g^b=gi,d^c=di

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, 405293);