#178222

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
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

615 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85418);