#179210

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86406);