#174823

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,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, 82019);