#152201

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

308 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^c=ei,f^c=fi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59397);