#200422

order	512
nilpotency class	3
isoclinism	isoclinic class of #200418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1356 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107618);