#289122

order	512
nilpotency class	3
isoclinism	isoclinic class of #289115
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

592 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:

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

Polycyclic presentation:

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

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, 196318);