#289106

order	512
nilpotency class	3
isoclinism	isoclinic class of #289075
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

578 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=i,d^2=ehi,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,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, 196302);