#236414

order	512
nilpotency class	3
isoclinism	isoclinic class of #236399
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	#60801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1256 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,41)(31,40)(33,39)(35,42)(37,43)(38,46)(44,45)(47,48);(4,5)(8,14)(10,16)(11,13)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,41)(26,29)(27,39)(28,40)(34,44)(35,45)(36,47)(37,46)(38,42)(43,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,15)(10,14)(17,20,23,37)(18,25,28,42)(19,30,33,46)(21,36,22,35)(24,38,40,41)(26,43,27,34)(29,44,39,48)(31,47,32,45);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,40,39)(25,34,42,43)(30,45,46,47)(38,48,41,44)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=h,f^2=hi,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,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143610);