#174811

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

631 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=ghi,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, 82007);