#179361

order	512
nilpotency class	3
isoclinism	isoclinic class of #179229
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 56 points:

(5,8)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24)(25,26,30,34)(27,35,37,48)(28,32,39,46)(29,41,43,33)(31,42,45,38)(36,49,51,56)(40,53,52,47)(44,54,55,50);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,23)(20,24)(25,27,30,37)(26,31,34,45)(28,36,39,51)(29,42,43,38)(32,44,46,55)(33,48,41,35)(40,54,52,50)(47,56,53,49);(1,3,4,7)(2,5,6,8)(9,11,13,18)(10,14,15,22)(12,20,19,17)(16,24,23,21)(25,28)(26,32)(27,36)(29,40)(30,39)(31,44)(33,47)(34,46)(35,49)(37,51)(38,50)(41,53)(42,54)(43,52)(45,55)(48,56);(10,15)(14,22)(16,23)(21,24)(27,37)(28,39)(31,45)(32,46)(35,48)(38,42)(40,52)(47,53)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,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, 86557);