#174812

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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 82008);