#170308

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

651 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)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18)(19,26)(20,23)(21,25)(22,31)(24,29)(27,28)(30,32)(33,34)(35,42)(36,40)(37,45)(38,41)(39,43)(44,47)(46,48);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,18)(19,27,30,24)(21,31,22,25)(26,28,32,29)(33,35)(34,39)(36,43)(37,46)(38,44)(40,47)(41,48)(42,45);(2,9)(3,12)(8,13)(11,14)(17,21,23,22)(18,25,20,31)(19,28,30,29)(24,26,27,32)(33,36,38,37)(34,40,41,45)(35,43,44,46)(39,47,48,42)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,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=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77504);