#151880

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₄²

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	C₂³	C₂×M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,21,15,23,38,54,16,24,39,55,35,50,64,71,17,22,40,53,33,48,63,70,13,20,36,51,12,19,34,49)(11,25,28,56,29,47,60,69,30,18,41,43,26,45,57,68,31,46,37,65,58,42,66,67,27,44,59,52,32,61,62,72);(2,5)(6,8)(9,11)(10,18,22,44)(12,32)(13,27)(14,37)(15,29)(16,30)(17,31)(19,47,50,42)(20,25,24,46)(21,52,53,43)(23,45,48,61)(26,35)(28,40)(33,58)(34,57)(36,41)(38,66)(39,59)(49,67,71,69)(51,65,55,56)(54,72,70,68)(60,63)(62,64);(1,3,4,7)(2,6,5,8)(9,12,13,33,17,35,16,15)(10,19,20,48,22,50,24,23)(11,26,27,29,31,32,30,58)(14,34,36,63,40,64,39,38)(18,42,25,45,44,47,46,61)(21,49,51,70,53,71,55,54)(28,57,59,60,37,62,41,66)(43,67,56,68,52,69,65,72)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59076);