#174614

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

640 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=gh,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, 81810);