#178718

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
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

598 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85914);