#169605

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
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

729 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,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, 76801);