#174826

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

679 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 82022);