#151756

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

392 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)(5,6,9,15,10,17,28,42,11,18,29,43,31,45,56,65,12,16,30,41,32,44,57,64,8,14,25,38,26,39,53,62)(7,19,22,46,23,47,51,66,24,13,33,35,52,36,58,60,20,37,27,54,48,61,55,68,21,34,49,40,50,59,67,63);(2,4)(5,7)(6,13,16,34)(8,21)(9,27)(10,23)(11,24)(12,20)(14,19,18,37)(15,40,41,35)(17,36,44,59)(22,30)(25,33)(26,50)(28,55)(29,49)(31,52)(32,48)(38,54,43,46)(39,47,45,61)(42,63,64,60)(51,57)(53,58)(56,67)(62,68,65,66);(7,20)(13,34)(19,37)(21,24)(22,27)(23,48)(33,49)(35,40)(36,59)(46,54)(47,61)(50,52)(51,55)(58,67)(60,63)(66,68)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322699, #72323051, #72323068
maximal normal subgroup of	#72322699, #72323051, #72323068
maximal quotient of	#72322703

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58952);