#153574

order	512
nilpotency class	6
isoclinism	isoclinic class of #13742
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₃₂
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	C₂³	D₃₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,3,4)(6,12)(7,13)(8,18)(9,22)(10,25)(14,27)(15,39)(16,42)(17,43)(19,47)(20,50)(21,35)(23,52)(24,46)(26,51)(28,29)(30,40)(31,38)(32,33)(34,48)(36,61)(37,63)(41,66)(44,45)(49,68)(53,54)(55,62)(56,65)(57,64)(58,67)(59,60);(5,6,13,29,8,15,32,42,22,40,47,65,45,57,20,31,10,17,34,58,23,41,49,55,26,37,60,61,46,53,21,27,11,14,35,54,24,36,59,63,51,62,68,66,52,67,48,43,25,38,50,64,44,56,19,30,9,16,33,39,18,28,7,12);(6,14)(12,27)(15,36)(16,37)(17,38)(28,53)(29,54)(30,55)(31,43)(39,61)(40,62)(41,56)(42,63)(57,67)(58,64)(65,66)

Polycyclic presentation:

2^9|a^2=e,b^2=df,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60770);