#151776

order	512
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	C₂⁴	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,f^2=g,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gh,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58972);