#236497

order	512
nilpotency class	3
isoclinism	isoclinic class of #236489
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,28)(20,25)(21,36)(22,26)(23,27)(24,30)(29,45)(31,39)(32,44)(33,38)(34,41)(35,43)(37,42)(40,46)(47,48);(4,13)(7,12)(9,16)(11,14)(17,19,23,32)(18,24,27,39)(20,33,34,47)(21,31,22,30)(25,40,41,45)(26,44,36,28)(29,37,46,35)(38,42,48,43);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,20,23,34)(18,25,27,41)(19,29,32,46)(21,35,22,37)(24,38,39,48)(26,42,36,43)(28,45,44,40)(30,33,31,47);(3,10)(6,15)(9,16)(11,14)(20,34)(25,41)(29,46)(33,47)(35,37)(38,48)(40,45)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,e^c=ei,f^c=fi,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, 143693);