#237061

order	512
nilpotency class	3
isoclinism	isoclinic class of #237043
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

1281 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)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,29)(22,31)(23,33)(25,32)(27,40)(28,36)(30,35)(34,37)(38,39);(2,7)(3,8)(4,6)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,21)(19,32)(20,33)(22,28)(26,39)(29,38)(30,37)(31,35)(36,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,26)(19,35)(20,36)(21,34)(23,37)(24,27)(25,38)(32,40)(33,39);(1,4,5,6)(2,7,3,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,33,32)(17,26,36,35)(22,37,38,27)(31,34,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144257);