#237339

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	#60965	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1384 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=i,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, 144535);