#220674

order	512
nilpotency class	3
isoclinism	isoclinic class of #220548
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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)(9,10,14,18,15,20,13,19)(11,21,24,16,26,32,25,22)(12,17,28,35,30,36,29,34)(23,37,39,31,40,33,27,38);(9,11,15,26)(10,16,20,22)(12,27,30,39)(13,25,14,24)(17,33,36,37)(18,32,19,21)(23,29,40,28)(31,35,38,34);(2,5)(3,7)(9,10,15,20)(11,22,26,16)(12,17,30,36)(13,18,14,19)(21,24,32,25)(23,38,40,31)(27,37,39,33)(28,34,29,35);(9,12)(10,17)(11,23)(13,29)(14,28)(15,30)(16,31)(18,35)(19,34)(20,36)(21,37)(22,38)(24,39)(25,27)(26,40)(32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127870);