#179378

order	512
nilpotency class	3
isoclinism	isoclinic class of #179229
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

604 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86574);