#237342

order	512
nilpotency class	3
isoclinism	isoclinic class of #237325
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60415	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1387 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144538);