#170520

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
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

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77716);