#179060

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

647 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:

(2,6)(3,4)(7,8)(10,15)(12,13)(16,23)(17,21)(18,19)(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);(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,22)(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);(1,2)(3,6)(4,8)(5,7)(9,11,14,20)(10,16,17,24)(12,19,13,18)(15,22,21,23)(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);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,15,17,21)(11,18,20,19)(16,23,24,22)(27,37)(28,39)(31,45)(32,46)(35,48)(38,42)(40,52)(47,53)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=hi,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, 86256);