#178990

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

652 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)(26,31)(28,36)(29,42)(30,37)(32,44)(33,48)(34,45)(35,41)(38,43)(39,51)(40,54)(46,55)(47,56)(49,53)(50,52);(1,2)(3,6)(4,8)(5,7)(9,11,14,20)(10,16,17,24)(12,19,13,18)(15,22,21,23)(25,28,30,39)(26,32,34,46)(27,36,37,51)(29,40,43,52)(31,44,45,55)(33,47,41,53)(35,49,48,56)(38,50,42,54);(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,c^2=g,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, 86186);