#289984

order	512
nilpotency class	3
isoclinism	isoclinic class of #289947
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

586 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:

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

Polycyclic presentation:

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

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, 197180);