#238971

order	512
nilpotency class	3
isoclinism	isoclinic class of #238970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

803 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)(4,14)(5,8)(6,9)(7,12)(11,16)(13,15)(17,18)(19,28)(20,33)(21,38)(22,27)(23,26)(24,30)(25,36)(29,44)(31,40)(32,43)(34,39)(35,42)(37,47)(41,46)(45,48);(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,19,23,32)(18,24,26,40)(20,34,36,45)(21,31,22,30)(25,41,33,44)(27,43,38,28)(29,35,46,37)(39,47,48,42);(1,3)(2,7)(4,13)(5,12)(6,11)(8,16)(9,15)(10,14)(17,20,21,35,23,36,22,37)(18,25,27,42,26,33,38,47)(19,29,30,45,32,46,31,34)(24,39,28,41,40,48,43,44);(3,11)(7,15)(10,16)(12,13)(18,26)(24,40)(25,33)(27,38)(28,43)(39,48)(41,44)(42,47)

Polycyclic presentation:

2^9|b^2=h,c^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=ghi,d^c=di,f^c=fhi,g^c=gi

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, 146167);