#174820

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
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	#2733	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

702 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 82016);