#239008

order	512
nilpotency class	3
isoclinism	isoclinic class of #239000
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

804 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)(3,7)(4,8)(5,9)(6,10)(11,15)(12,14)(13,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,37)(30,45)(31,40)(33,41)(35,48)(36,44)(38,43)(39,46)(42,47);(1,3)(2,7)(4,12)(5,11)(6,13)(8,14)(9,15)(10,16)(17,19,23,33)(18,24,28,40)(20,35,37,46)(21,31,22,32)(25,42,34,45)(26,29,27,41)(30,36,47,38)(39,44,48,43);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(17,20,21,36,23,37,22,38)(18,25,27,44,28,34,26,43)(19,30,32,46,33,47,31,35)(24,39,29,45,40,48,41,42);(1,4,6,5)(2,8,10,9)(3,12,13,11)(7,14,16,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,29,40,41)(25,43,34,44)(30,35,47,46)(39,45,48,42)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146204);