#237077

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1206 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,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, 144273);