#237476

order	512
nilpotency class	3
isoclinism	isoclinic class of #237463
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	#60962	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1292 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,e^c=ei,f^c=fhi,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, 144672);